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S  PEER'S 


Arithmetics 


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IN  MEMORIAM 
FLORIAN  CAJORI 


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ADVANCED  ARITHMETIC 


BY 

WILLIAM   W.   SPEER 

DISTRICT  SUPERINTENDENT  OF  SCHOOLS,  CHICAGO 


"  This  law  of  organic  progress  is  the  law  of  all  progress.  Whether 
it  be  in  the  development  of  the  earth,  in  the  development  of  life 
upon  its  surface,  in  the  development  of  society,  of  government,  of 
manufactures,  of  commerce,  of  language,  literature,  science,  art,  this 
same  evolution  of  the  simple  into  the  complex,  through  a  process  of 
continuous  diiTerentiation,  holds  throughout." 

—  Herbert  Spencer. 


BOSTON,  U.S.A. 
GINN   &   COMPANY,   PUBLISHERS 

C!)e  ^tjenaettm  press 
.    1899 


Copyright,  1899 
By  WILLIAM  W.  SPEER 


ALL  RIGHTS  RESERVED 


PREFACE 


The  purpose  of  this  book  is  to  aid  the  teacher  in  making 
conditions  favorable  for  the  contact  of  the  learner  with 
mathematical  realities.  Since  the  simplest  as  well  as  the 
most  complex  relations  become  known  only  through  mental 
activity  in  comparing,  attention  is  given  to  an  environment 
which  shall  continually  induce  this  activity.  The  compari- 
son of  magnitudes  includes  all  the  other  operations  of  arith- 
metic. This  is  apparent  when  we  reflect  that  all  advance 
in  knowing  is  by  progressive  acts  of  analysis  and  synthesis. 
Whether  we  compare  by  means  of  the  object  or  by  means  of 
its  symbol,  the  mind  should  still  be  free  to  move  from  the 
vague  to  the  definite  through  its  own  acts. 

In  the  first  book  of  this  series  it  was  said :  "  That  quan- 
tity is  a  ratio  between  terms  which  are  themselves  relative 
is  a  truth  which  has  often  been  pointed  out,  but  which  the 
work  of  the  schools  shows  to  be  felt  by  few."  In  the  minds 
of  many  bred  upon  the  language  of  mathematics,  the  mode 
of  expression  has  acquired  an  independence  which  excludes 
the  reality.  I'he  presentation  of  relative  magnitude  as  a 
subject  of  study  in  the  elementary  school  has,  however,  met 
with  gratifying  response.  Some  authors,  indeed,  whose 
presentations  are  entirely  foreign  to  the  development  of 
ideas  of  relative  magnitude,  have  recently  written  urging 

iii 


iv  PREFACE. 

relations  of  magnitude  as  the  objects  of  study  ;  others  have 
given  the  word  ''■  ratio  "  a  prominent  place  in  new  editions. 
"  A  religious  vocabulary  without  religious  experiences '  ■  is 
of  little  value  ;  a  mathematical  vocabulary  without  mathe- 
matical experiences  is  of  no  more  value.  The  student  can- 
not advance  in  any  science  if  his  attention  is  absorbed  in 
the  language.  Non-mathematical  work  cannot  generate 
mathematical  ideas.  Definitions  of  mathematics  and  of 
quantity  avail  little  without  the  feeling  and  the  intimate 
understanding  which  cause  us  to  shape  oui'  work  in  accord 
with  the  illuminating  idea. 

The  mode  of  dealing  with  the  greatest  common  measure, 
percentage,  longitude  and  time,  square  root,  mensuration 
is  in  accord  with  underlying  mathematical  ideas.  The 
pupil  advances  in  the  indirect  establishment  of  relations  of 
magnitude  by  living  in  practical  contact  with  mathematical 
realities.^ 

Each  subject  has  its  own  elementary  ideas.  The  study 
of  relative  magnitude  does  not  give  a  basis  for  inferences  in 
history  or  biology,  nor  does  attention  to  history  give  a  basis 
for  mathematics.  The  power  to  say,  "  The  act  is  brave,"  or, 
"  The  building  is  fitly  proportioned,"  begins  in  comparing 
through  the  senses,  but  the  observing  which  leads  to  one 
of  these  judgments  does  not  furnish  a  basis  for  the  other. 
But  all  elementary  work  should  be  formative  of  a  mental 
habit  which  may  be  fitly  carried  into  all  lines  of  effort. 

1  "  One  well-known  principle  underlying  the  acquisition  of  knowl- 
edge is  that  an  idea  cannot  be  fully  grasped,  by  the  youthful  mind 
unless  it  is  presented  under  a  concrete  form.  Whenever  possible  an 
abstract  idea  must  be  embodied  in  some  visible  representation.  .  .  . 
Should  it  appear  to  any  one  that  we  thus  detract  from  the  generality 
of  algebraic  quantities,  it  is  sufficient  to  reply  that  the  system  is  the 
same  which  mathematicians  use  to  assist  their  conceptions  of  advanced 
algebra,  and  without  which  they  would  never  have  been  able  to  grasp 
the  complicated  relations  of  imaginary  quantities. ' ' —  Simon  Newcomb. 


PBEFACB.  V 

Things  must  be  observed  in  various  aspects  if  they  are 
to  be  known,  but  to  say  that  too  much  attention  to  mathe- 
matics or  to  particular  things  weakens  is  merely  to  make  a 
specific  application  of  a  universal  truth. ^  The  presentation 
in  this  series  is  from  the  standpoint  that  any  success  is 
dangerous  which  lessens  the  susceptibility  of  the  mind. 
The  constant  purpose  is  to  promote  growing  power  to  act 

in  new  circumstances. 

W.  W.  SPEER. 

1  "The  intensity  with  which  any  form  of  exercise  is  carried  on 
during  the  growing  period  leaves  its  trace,  and  the  absence  of  it  at 
the  proper  time  is  for  the  most  part  irremediable.  We  should  hardly 
expect  much  appreciation  of  color  in  a  person  brought  up  in  the  dark, 
however  good  his  natural  endowments  in  this  direction.  Thus  any 
lack  of  early  experience  may  leave  a  spot  permanently  undeveloped 
in  the  central  system." — Prof.  H.  H.  Donaldson. 

Nor  can  we  expect  appreciation  of  mathematical  relations  if  there 
is  no  opportunity  for  attending  to  them.  If  we  leave  the  sense  of 
proportion  undeveloped,  and  leave  the  pupil  unaware  of  the  realities 
of  mathematics  in  the  elementary  work,  we  can  scarcely  expect  inter- 
est in  relative  magnitude  at  a  later  period.  —  Author. 


INTEODUCTIOJ^. 


Interdependence  of  the  Powers.  —  At  all  stages  education 
involves  the  whole  personality.  Continued  impressibility 
is  a  condition  of  growing  power  to  think  and  act.  Attempts 
to  substitute  the  analytic-synthetic  acts  of  the  adult  for 
those  of  the  child  separate  the  child  from  realities  of  which 
he  would  easily  possess  himself  if  not  interfered  with. 
Through  many  facts  we  rise  to  laws  which  embrace  them 
all  and  lead  to  other  truths.  So  the  higher  mental  life 
embraces  the  lower  and  further  develops,  not  by  disuse  of 
the  powers,  but  by  increasingly  effective  use  of  them. 
Varied  se/f-activity  is  the  great  characteristic  which  dis- 
tinguishes higher  from  lower  organisms. 

Means  and  End.  —  Intelligent  effort  is  the  adaptation  of 
means  to  end.  Mathematics  is  a  means  of  educating,  just 
as  botany  and  history  are.  From  the  mathematical  side 
the  purpose  of  the  study  is  the  discovery  of  the  relative 
magnitude  of  things.  What,  then,  should  be  the  course  of 
procedure  ?  Manifestly  the  learner  cannot  respond  to  that 
by  which  he  is  not  affected.  Is  there  any  way  by  which  he 
can  become  affected  by  relations  of  magnitude  except  by  his 
own  activity  ^  of  sense  and  mind  in  regard  to  such  relations? 

1  "Thus  even  when  the  relation  is  directly  presented,  e.gr.,  in  the 
spatial  relation  of  two  simultaneously  perceived  objects,  it  is  evident 
that  attention  must  direct  itself  to  this  relation,  and  selectively  bring 
it  into  mental  prominence." — James  Sully. 

vii 


Vm  INTRODUCTION. 

If  the  self-activity  which  gives  imperfectly  quantitative 
ideas  is  checked,  sensitiveness  to  mathematical  relations  is 
not  developed,  and  the  manipulative  operations  the  student 
is  called  on  to  perform  in  the  name  of  mathematics  become 
a  barrier  to  the  perception  of  mathematical  relations.  This 
activity  is  checked  when  appliances  are  so  pressed  upon  the 
attention  that  relative  magnitude  is  prevented  from  rising 
into  the  foreground  of  consciousness.  It  is  checked  when- 
ever methods  do  away  with  the  need  of  forming  personal 
judgments.  ' 

It  is  generally  admitted  that  we  cannot  know  a  thing  in 
isolation.  If  we  say,  "  The  day  is  warm,"  "  The  ratio  of  this 
line  to  that  is  3,"  '^  The  color  of  this  leaf  is  darker  than 
that,"  or,  "This  is  a  dog,"  we  have  expressed  the  likeness  or 
unlikeness  of  things.  It  is  also  recognized  that  to  establish  a 
relation  between  two  things  the  things  must  be  present  in 
consciousness.  The  mode  by  which  the  mind  enters  into 
the  various  aspects  of  truth  is  fundamentally  the  same ;  it 
is  by  thinking.  And  what  is  thinking  ?  We  talk  freely 
of  training  children  to  think,  but  what  do  we  mean  ?  How 
do  we  set  about  it  ?  Is  thinking  something  separate  and 
apart,  something  carried  on  independently  of  sensing  and 
feeling  ?  Or  are  the  things  we  sense  and  feel  the  things 
we  relate  ?  Can  relations  be  discovered  without  the  forma- 
tion of  judgments?  Can  a  relation  be  discovered  unless 
there  are  at  least  two  terms  ?  Why  not  ?  Can  we  dis- 
criminate, classify,  form  judgments,  see  likenesses  and 
differences  unless  we  compare  ? 

From  Plato  down,  relating  has  been  held  to  be  the  essence 
of  the  intellectual  life.  Why  should  we  shun  this  intel- 
lectual act,  or  fail  to  foster  it  in  mathematics  more  than 
in  any  other  subject  ? 

The  Mental  Whole.  —  Whatever  the  mind  occupies  itself 
with  becomes  a  whole,  the  object  of  thought  for  the  time 


INTRODUCTION.  IX 

being. ^  To  think  of  the  size  of  a  house  means  to  think  of 
the  relation  of  magnitude  which  it  as  a  whole  bears  to 
other  wholes.  To  think  of  relative  times,  weights,  values, 
etc.,  means  to  think  of  each  of  the  wholes  in  the  particular 
relation  specified.-  Any  presentation  which  interferes  with 
the  mental  grasp  of  the  terms  of  comparison  as  units  inter- 
feres with  the  perception  of  relative  magnitude. 

The  Mathematical  Element ;  Coexistence  ;  Coextension.  — 
That  certain  conditions  foster  and  others  interfere  with 
growth  into  mathematical  ideas  may  become  more  clear 
when  we  consider  what  is  involved  in  the  perception  of 
a  quantitative  relation. 

The  simplest  comparison  implies  coexistent  wholes.  The 
object  is  to  discover  the  relative  magnitude  of  the  wholes 
under  consideration.  All  intermediate  processes,  all  the 
separating  and  combining,  have  this  end  in  view.  It  is 
clear,  then,  that  each  unit  must  be  grasped  as  a  whole. 
Now  the  use  of  the  discontinuous  unit  directs  attention  not 
to  one  unit  but  to  several.  When  a  child's  attention  is 
centered  upon  the  parts,  as  when  he  begins  with  counting 
or  when  he  applies  an  artificial  measure  before  making  any 
mental  note  of  a  whole,  each  separately  apprehended  part 
becomes  a  whole ;  consequently,  the  units  which  should  be 
compared  are  not  imaged  and  not  compared.  The  expression 
of  their  relation  is  then  a  formal  act,  not  an  expression  of 
the  perception  of  their  relative  magnitude.  A  quantity 
becomes  discrete  for  an  individual  only  when  he  makes  it 
so  by  his  own  intellectual  acts,  his  'own  analyzing.^  A 
synthesis  not  accompanied  by  an  analysis  is  artificial.     If 

1  See  "  Teachers'  Book,"  p.  17. 

2  Whether  a  relation  is  to  be  expressed  in  the  language  of  percentage 
or  in  some  other  manner,  the  pupil  should  still  be  given  opportunity 
to  perceive  the  units  he  is  to  compare  as  coexisting  and  coextensive. 

3  See  "Teachers'  Book,"  p.  18. 


INTRODUCTION. 


we  wish  the  terms r to  be  felt,  is  it  not  a  mis- 


take to  present  them  broken  into  parts,  or  discontinuous  ? 
By  so  doing  we  prevent  the  mind  from  grasping  the  terms 
which  should  be  compared,  and  therefore  from  grasping 
their  relation. 

If  we  ask  for  the  relative  magnitude  of  a  and  6,  we  tacitly 
assert  the  coexistence  of  the  magnitudes.  Why,  then,  should 
we  not  present  them  as  coexistent  ?  In  Mr.  Spencer's  words, 
"Coexistence  implies  two  somethings  that  coexist.  Two 
somethings  cannot  occupy  the  same  space, .  .  .  hence  things 
cannot  be  known  as  coexistent  without  also  being  known 
as  out  of  each  other  —  at  some  distance  from  each  other." 
What  does  this  suggest  ?  The  presentation  should  foster 
the  perception  of  the  coextension  of  coexistent  magnitudes 
of  like  kind.  Without  such  perceptions  there  is  no  quan- 
titative reasoning.^  Upon  these  perceptions  the  construc- 
tions of  mathematics  rest. 

Rhythm;  Counting. — Consciousness  consists  of  successive 
states.  Rhythm  is  a  factor  in  life.  But  these  truisms  have 
no  more  to  do  with  relative  magnitude  as  the  object  of 
mathematical  study  than  with  the  ideas  which  characterize 
chemistry  or  botany.  That  the  relation  of  coexistence  is 
not  directly  grasped  in  the  first  stages  of  development,  but  is 

1  "  Thus,  abstract  as  they  have  now  become,  the  units  of  calculation, 
applied  to  whatever  species  of  magnitudes,  do  really  stand  for  equal 
units  of  linear  extension ;  and  the  idea  of  coextension  underlies  every 
process  of  mathematical  analysis.  Similarly  with  coexistence.  Numer- 
ical *  symbols  are,  it  is  true,  purely  representative  ;  and  hence  may  be 
regarded  as  having  nothing  but  a  fictitious  existence.  .  .  .  The  cal- 
culus, in  general,  is  a  means  of  dealing  with  magnitudes  that  do  not 
coexist,  or  are  not  homogeneous,  or  both,  by  first  substituting  for  them 
magnitudes  that  do  coexist  and  are  homogeneous,  and  afterwards 
re-translating  these  into  their  original  forms." — Herbert  Spencer. 

*  Numbers.    See  "  Principles  of  Psychology,"  vol.  ii,  pp.  38,  39. 


INTRODUCTION.  XI 

arrived  at  by  experiences  in  relating,  is  an  additional  reason 
for  conditions  which  elicit  activity  in  comparing.  We 
have  chemical,  astronomical,  and  biologic  rhythms ;  but  we 
do  not,  therefore,  begin  these  subjects  with  exercises  in 
counting.  Whether  comparing  things  in  respect  to  color, 
beauty,  taste,  or  magnitude,  the  mind  moves  from  the  whole 
it  grasps  to  the  parts.  If  we  wish  the  child  to  see  rela- 
tions of  likeness  and  unlikeness  between  an  orange  and  an 
apple,  we  bring  the  orange  and  the  apple  before  him.  We 
do  not  put  parts  before  him  and  ask  him  to  look  at  them. 
Why  not  pursue  a  similar  course  if  we  wish  him  to  see  the 
likeness  or  difference  in  the  magnitude  of  things  ? 

If  we  believe  the  appreciation  of  a  picture  or  a  poem 
comes  through  absorbing  the  mind  in  details  before  the 
whole  has  been  seen,  if  we  believe  the  child  likes  to  act 
upon  elements  which  the  adult  has  dissociated  from  the 
concrete  whole,  then,  naturally  we  make  counting  the  basis 
in  mathematics.  But  if  we  believe  that  in  all  thinking, 
whatever  the  mind  attends  to  becomes  the  whole,  that  it 
can  relate  only  what  it  grasps,  we  naturally  present  coex- 
istent, coextensive  magnitudes.  Analyzing  for  himself, 
the  learner  does  not  lose  his  grasp  of  the  things  to  be  com- 
pared. He  advances  by  progressive  analysis  and  synthesis 
according  to  his  capacity. 

Self-Activity  the  Means  of  Advance.  —  Progress  in  any 
science  implies  growth.  Growth  depends  upon  self-activity. 
Those  who  accept  these  ideas  cannot  consistently  substitute 
analyses,  ruljes,  or  mechanical  measuring  for  the  self -activity 
through  which  the  mind  grows  into  mathematical  ideas. 
I  say  "  grows "  because  only  through  this  evolutionary 
process  can  there  arise  an  inner  I'elation  answering  to  an 
outer  one ;  only  thus  can  objective  truth  be  realized. 
Growth  into  mathematical,  as  into  other  ideas,  is  the, slow, 
gradual  result  of  continuous  judging,  and  an  environment 


Xii  INTRODUCTION. 

which  continually  excites  such  activity  is  the  means  of 
teaching  mathematics.  This  activity  in  judging  of  relative 
magnitude  no  more  involves  mental  strain  than  the  activity 
expressed  in  the  judgment,  "The  book  is  green,"  "This 
candy  is  sweeter  than  that,"  or  "  The  boy  is  good."  All 
these  ideas  are  the  result  of  individual  activity  in  comparing. 

Digressing  slightly,  I  would  call  the  attention  of  teachers 
to  the  fact  that  expressions,  even  though  they  are  the  genuine 
products  of  mental  action  on  the  part  of  the  child,  may  not 
signify  to  him  what  they  do  to  the  adult.  They  can  do  so 
only  through  many  and  varied  experiences.-^  "  The  boy  is 
good,"  "  The  ratio  of  8  to  4  is  2,"  are  ideas  which  become 
more  complete  and  definite  with  developing  experiences.^ 

Through  Indefinite  to  Definite.  —  Mathematics  offers  no 
exception  to  the  rule  that  the  mind  advances  from  the 
indefinite  to  the  definite  by  its  own  acts.  The  mental 
process  by  which  the  child  becomes  aware  of  marked  differ- 
ences in  color  or  magnitude  is  the  only  process  by  which 
he  can  become  aware  of  finer  distinctions  and  closer  like- 
nesses. Eealization  of  this  forbids  attempts  to  create 
definite  ideas  by  a  use  of  mechanical  apparatus  which 
subordinates  mental  action. 

In  the  first  book  of  this  series  attention  was  called  to  the 
essential  likeness  of  reasoning  in  mathematics  and  in  other 
subjects.  In  the  words  of  Mr.  John  Fiske  :  "  Between  the 
two  kinds  of  reasoning  the  only  essential  difference  is  the 
degree  of  definiteness  to  which  they  are  respectively  devel- 
oped."    And  Mr.  Herbert  Spencer  has  pointed  out  that^ 

1  See  "Teachers'  Book,"  p.  15. 

2  A  child  of  four,  reading,  "  Tom  has  a  knife.  It  is  a  good  knife," 
said  with  surprise  :  "  What*  makes  them  say  that  ?  Knives  can't  be 
naughty,  can  they  ?  "  This  suggests  the  varied  experiences  needed  to 
give  meaning  to  abstract  and  general  terms. 

3  "A  man  who  has  walked  a  mile  in  fifteen  minutes,  and,  observing 
that  he  has  a  quarter  of  a  mile  still  to  go,  infers  the  time  it  will  take 


INTRODUCTION.  XIU 

"Every  act  of  quantitative  reasoning  is  qualitative  in  its 
initial  stages."  That  is,  the  mind  grasps  things  vaguely  as 
wholes  and  relates  them  as  more  and  less  before  it  arrives 
at  their  exact  relation  by  the  aid  of  ai'tifices  of  calculation. 
The  mental  act  may  be  so  rapid  that  ordinarily  we  fail  to 
note  it,  but  progress  is  through  the  free  movement  of  the 
self-active  mind.  The  power  to  establish  definite  relations 
is  but  a  higher  development  of  the  power  to  establish 
indefinite  relations. 

Environment.  —  The  environment  of  the  school  includes 
the  material,  the  demands,  methods ;  in  short,  all  to  which 
the  child  conforms  himself. 

As  implied  above,  growth  into  mathematical  ideas  is  not 
the  result  of  using  definite  measures  or  definite  language, 
but  of  continued  contact  with  relative  magnitude.  There 
is  no  such  contact  unless  the  mind  brings  things  into  quan- 
titative comparison.  It  cannot  do  so  if  energy  is  engrossed 
in  something  else,  —  if  a  growing  sensitiveness  to  mathe- 
matical realities  is  prevented  by  formal  work,  by  absorption 
in  appliances  for  securing  outside  products.  As  the  school 
aids  the  evolution  of  ideas  in  botany  or  history  by  condi- 
tions which  promote  growing  consciousness  of  these  phases 
of  truth,  so  it  should  do  in  mathematics.  Allowing  for 
natural  aptitudes  and  capacity,  the  child  attends  to  that  in 
which  it  sees  others  interested.  The  impressions  we  receive 
are  the  cause  of  our  reactions.  If,  then,  the  school  places  a 
fictitious  value  upon  outside  eft'ects  and  limits  sensory  and 
motor  reaction  to  the  stimuli  of  mechanical  operations,  the 
experiences  which  cause  the  progressive  development  of  the 
mind  do  not  take  place.    By  mistaking  the  means  for  the  end 

to  reach  his  destination,  does  not  primarily  infer  three  minutes  and 
three-quarters^  he  primarily  infers  a  short  time  —  a  time  indefinitely 
conceived  as  less  than  ten  minutes  and  more  than  one." — Herbert 
Spencer. 


XIV  INTRODUCTION. 

we  too  often  shut  the  child  out  from  these  experiences.  As 
freedom  may  be  lost  by  exalting  special  forms  of  free  gov- 
ernment, so  mathematics  may  be  obscured  by  its  appliances.^ 
The  measuring  which  develops  mathematical  ideas  is  not  a 
series  of  mechanical  operations  akin  to  the  automatic  prac- 
tice of  the  carpenter  with  his  rule,  but  the  mental  compar- 
ing, without  which  things,  as  we  know  them,  do  not  exist 
for  us.  The  so-called  objective  method  may  no  more  ignore 
the  perceiving  mind  than  the  mind  may  ignore  the  continual 
verification  of  experiment. 

The  learner  may  make  due  use  of  a  measure  in  testing 
a  judgment  after  he  has  made  the  judgment.  To  supple- 
ment the  activity  of  mind  and  senses  is  one  thing,  to  super- 
sede it  is  another.  We  check  development  and  rob  the 
pupil  of  the  experiences  which  expand  the  idea  and  the 
mind,  when  we  limit  energy  to  mechanical  apparatus  or 
to  processes  for  securing  particular  results. 

"The  study  of  mathematics,"  said  Professor  Sylvester, 
"is  unceasingly  calling  forth  the  faculties  of  observation 
and  comparison."  By  examples  seemingly  far  removed  from 
sense  he  showed  that  mathematical  discovery  is  always  the 
outgrowth  of  sensory  and  intellectual  activity.  Eeliance 
upon  artificial  measuring,  or  that  vicarious  analyzing  which 
presents  the  terms  already  divided,  is  but  a  variation  of  the 
old  plan  of  getting  results  by  rule.  Why  should  a  child 
think  when  a  rule  will  serve  ?  Why  should  he  compare  the 
magnitude  of  things,  why  judge  of  their  relative  magnitude, 
when  a  foot  rule,  an  inch,  or  a  pound  weight  will  do  away 
with  the  need  of  mental  movement  ?  Why  should  he  look 
at  plants  when  the  catalogue  of  the  naturalist  is  ready  to 

1  "All  the  devices  familiar  to  the  reader  of  Euclid  .  .  .  are  simply 
devices  for  bringing  a  given  pair  of  space  relations  directly  into  con- 
sciousness, so  that  their  equality  or  inequality  may  be  recognized  by 
direct  inspection."— John  Fiske. 


INTRODUCTION.  XV 

his  hand  ?  Perhaps  for  no  reason  except  that  he  cannot 
advance  in  any  science,  or  in  the  power  of  definite  thinking, 
without  continued  mental  activity. 

The  school  should  economize  energy  by  a  presentation 
which  attracts  attention  to  the  essential.  It  should  reen- 
f orce  self-activity  by  an  environment  fitted  for  the  developing 
powers. 

Formal  Analyses. — Another  phase  of  counterfeit  activity 
in  mathematics  is  formal,  logical  analysis.  As  making 
and  measuring  by  rule  give  artificial  products,  so  do  analy- 
ses which  make  children  the  "  parrots  of  other  men's  think- 
ing." They  stifle  thought  and  conceal  the  condition  of  the 
learner.  Professor  Jevons  invented  a  logical  machine,  but 
he  did  not  claim  that  it  acted  intelligently.  Manipulating 
formal  analyses  may  satisfy  the  demands  of  the  school,  but 
not  of  life. 

The  repetition  of  the  adult's  "  sinces  "  and  "  theref ores  " 
is  not  reasoning,^  nor  is  mental  power  increased  by  requiring 
the  student  continually  to  set  forth  every  step  by  which  a 
conclusion  is  reached.  The  work  should  call  for  personal 
expression,  and  correct  decision  is  evidence  that  the  mind 
is  not  moving  amiss.  Why  explain  again  and  again  the 
reasons  for  the  inference  that  in  6  hours  we  may  walk  2  x 
miles,  if  in  3  hours  we  walk  x  miles,  any  more  than  for  the 
inference  that  the  apple  is  good  ?  The  child  is  not  called 
upon  to  analyze  the  acts  by  which  he  classifies  this  as  an 
orange,  that  as  a  pear.     It  is  sufficient  that  he  decides  cor- 

1  "The  doctrine  that  we  can  discover  facts,  detect  the  hidden  proc- 
esses of  nature,  by  an  artful  manipulation  of  language,  is  so  contrary 
to  common  sense  that  a  person  must  have  made  some  advances  in 
philosophy  to  believe  it." — Johx  Stuart  Mill. 

"  High  Air-castles  are  cunningly  built  of  Words,  the  Words  well 
bedded  also  in  good  Logic-mortar  ;  wherein,  however,  no  Knowledge 
will  come  to  lodge." — Thomas  Carlyle. 


XVI  INTRODUCTION. 

rectly.  If  bodily  and  intellectual  acts  did  not  sink  into  the 
automatic,  progress  would  soon  be  at  a  standstill. 

By  a  multitude  of  experiences  the  child  is  able  so  swiftly 
to  compare  two  pails,  and  to  infer  that  the  larger  will  hold  a 
larger  amount,  and  to  act  accordingly,  that  perception,  infer- 
ence, and  choice  seem  one ;  and  who  would  not  think  it 
absurd  to  require  verbal  proof  of  the  wisdom  of  the  choice  ? 
Mental  as  well  as  physical  experiences  become  organically 
registered,  and  the  mind  does  not  dwell  upon  familiar  per- 
ceptions any  more  than  it  does  upon  the  series  of  acts 
involved  in  reading  or  playing  the  piano.  The  mind  must 
analyze  in  order  to  read  a  page,  but  what  would  be  the  effect 
of  dwelling  upon  each  separate  word,  syllable,  and 
letter  ? 

There  is  no  more  reason  for  asking,  "How  do  you  see 
this  ?  ^'  in  mathematics  than  in  any  other  subject.  The 
result  of  attempts  to  apply  the  introspective  method  to 
adult  consciousness  does  not  recommend  this  plan  with 
children.  The  tendency  to  suspend  judgment  and  to  rest 
opinions  upon  facts  instead  of  upon  conj  ecture  is  not  the 
result  of  dwelling  upon  details.  The  demand  for  continually 
particularizing  every  step  by  which  we  reach  a  conclusion 
interferes  with  the  forward  reaching  functions  of  the  mind 
and  with  the  growing  power  to  separate  the  essential  from 
the  non-essential.  It  is  the  continued  adjustment  of  the 
mind  to  things  which  makes  their  connections  familiar  and 
necessary. 

Motor  Activity.  —  As  previously  urged,  methods  which 
separate  the  thinking  being  from  the  being  who  senses, 
feels,  and  acts  are  mischievous.  At  no  stage  can  we  afford 
to  neglect  the  motor  activity  which  promotes  free  circulation 
and  supplies  the  brain  with  the  blood  necessary  for  atten- 
tion and  achievement.  Moreover,  the  dependence  of  devel- 
oping intelligence  upon  developing  nervous  structure  urges 


INTBODUCTION.  XVll 

provision  for  sensory-motor  processes,  which  supply  nerve 
energy  and  excite  intellectual  responsiveness  in  given  direc- 
tions. 

There  is  no  reason  for  continuing  to  neglect  the  plain 
teaching  of  science  concerning  the  bearing  of  varied  move- 
ment upon  mental  nutrition.  Sensing  leads  not  only  to 
imagination  and  inference  but  to  outward  expression,  and 
the  school  should  open  the  way  for  the  inner  life  to  perfect 
itself  in  action. 

The  bearing  of  the  tactual  and  muscular  senses  upon 
development  of  motor  intuitions  and  upon  the  blending  of 
motor  intuitions  into  ideas  of  distance,  size,  etc.,  are  too 
familiar  to  need  repeating.  Adapted  movement  is  an 
important  factor  in  forming  and  fixing  mathematical  judg- 
ments, but  it  must  not  be  forgotten  that  there*  may  be 
activity  of  the  hand  as  well  as  of  the  tongue  without 
reflecting  the  mind,  —  activity  which  represses  instead,  of 
develops.  Motor  action  is  a  means  of  inducing  and  cor- 
recting needful  judgments  as  well  as  an  expression  of  what 
is  seen,  but  motor  action  in  drawing,  cutting,  making,  etc., 
should  be  the  genuine  expression  of  action  within  the 
doer.  Making  by  rule  and  measure  no  more  produces  a 
mental  structure  capable  of  picturing  conditions,  or  gives 
a  growing  sense  of  mathematical  relations,  than  does 
repeating  formulas  without  the  experiences  M^hich  give 
them  significance.^  Right  manual  training  means  mental 
training,  and  it  is  an  essential  factor  of  education.  We  do 
not  meet  the  needs  of  the  child  by  giving  him  a  rule  or  a 
tool  for  producing  a  particular  result.     That  which  is  with- 

1  "  The  intellectual  factor  of  manual  labor  is  never  very  large  even 
in  the  first  construction  of  a  new  type  of  product.  The  moral  educa- 
tion in  manual  training  in  the  way  of  perseverance,  patience  and 
plodding  industry  is  a  far  greater  educational  factor  than  the  intel- 
lectual factor."  —  W.  T.  Harris,  "Report  Com.  on  Pedagogy,"  1889. 


xviii  INTRODUCTION. 

out  and  that  which  is  within  is  brought  into  harmony  by 
action  which  is  the  outgrowth  of  perception,  imagery,  and 
will.  Effects  transcending  the  receptive,  analytic-synthetic 
power  of  the  mind  are  fatal  to  self-activity  and  to  intel- 
lectual and  moral  growth. 

A  child  may  apply  a  foot  rule  three  times  to  the  length 
of  a  box,  and  twice  to  its  width,  and  then  mark  off  the  same 
number  of  spaces  on  another  piece  of  wood  or  pasteboard, 
and,  by  a  series  of  similar  processes,  construct  articles  innu- 
merable with  scarcely  a  glimmering  of  proportion  or  a 
stirring  of  the  representative  power.  Year  after  year  the 
carpenter  measures  without  more  progress  in  mathematics 
than  the  farmer,  just  as  year  after  year  we  may  handle  the! 
crucibles  of  the  chemist  without  advancing  in  chemistry,  or' 
repeat  the  formulas  of  the  thinkers  without  thinking. 

The  equations  of  mathematics  express  real,  not  formal, 
truths ;  and  the  only  means  of  contact  with  these  truths  is 
through  the  activity  of  the  mind  in  regard  to  the  relative 
magnitude  of  things.  There  is  no  other  mode  of  adjustment. 
Newton  urged  that  if  ever  the  action  of  observant  senses 
and  mind  in  regard  to  the  mathematical  relation  of  things 
was  displaced  by  artifices  of  calculation,  then  would  the 
study  of  mathematics  degenerate  into  something  little  better 
than  handicraft  dexterity. 

Whenever  the  method  of  the  school  fails  to  preserve 
the  balance  between  impressibility  and  outer  activity,  when- 
ever it  centers  upon  outward  products,  whether  secured  by 
formulas  or  measures,  to  the  neglect  of  mental  receptivity, 
responsiveness,  and  growing  power,  it  is  non-developing. 

Concrete  and  Abstract.  —  Continued  activity  in  a  varied 

environment  ensures  progressive  abstraction.    Gradually  the 

mind  is  freed  from  the  concrete  and  the  particular.^     The 

power  to  think  in  general  and  abstract  terms  is  the  product 

1  See  "  Teachers'  Book,"  pp.  19,  20. 


TNTEODUCTION.  XIX 

of  many  experiences  through  which  the  conscious  becomes 
unconscious,  and  memories  become  organic.  As  from  par- 
ticular plants  and  minerals  we  rise  to  a  comprehension  of 
the  types  with  which  science  deals,  so  we  advance  in 
mathematics. 

No  definite  lines  can  be  drawn  between  the  power  of 
perceiving  and  the  power  of  inferring  relations,  nor  between 
the  power  to  compare  by  means  of  visible  things  and  the 
power  to  compare  relations  without  the  aid  of  the  things 
displaying  them.  If  we  attribute  to  the  childish  mind  the 
sense  of  proportion,  of  ratios,  which  belongs  to  the  adult, 
and  so  neglect  the  manifold  experiences,^  the  acts  of  asso- 
ciating and  dissociating  by  which  the  pupil  grows  into 
definite  ideas,  we  fail  to  teach  mathematics.  Every  fresh 
experience,  every  new  perception  of  the  concrete,  gives 
greater  power  to  think  in  general  and  abstract  terms. 

Not  merely  arithmetic,  but  the  science  of  mathematics  is 
a  whole,  and  in  the  elementary  work  a  bias  is  given  to  the 
mind  which  fits  or  unfits  it  for  discovery.  Not  by  dwell- 
ing in  turn  upon  the  language  and  devices  of  arithmetic, 
geometry,  and  algebra,  but  by  a  presentation  that  promotes 
those  unifying  processes  which  take  place  within  the  mind, 
do  we  promote  progress  in  mathematics.  Only  by  the  evo- 
lution of  interest,  action,  and  power  is  the  mind  prepared 
for  high  abstraction.  Stereotyping  upon  the  brain  isolated 
images  which  ^'  live  alone  and  die  alone  incapable  of  fecun- 
dation "  is  not  the  means  by  which  to  develop  that  repre- 
sentative power  which  is  the  mark  of  intellectual  progress. 
"  The  higher  processes  of  the  mind  in  mathematics  lie  at 
the  very  foundation  of  the  subject."     The  relations  arrived 

1  "  Men  fail  to  judge  correctly  because  they  have  not  had  that  suffi- 
cient instruction  by  the  senses  which  would  justify  making  a  conclu- 
sion. We  must  contrive  extra  and  special  means  by  which  their  first 
impressions  shall  be  corrected  and  enlarged." — Faraday. 


XX  INTRODUCTION. 

at  through  symbols  are  just  as  real  as  those  directly  dis- 
cerned through  things,  but  symbols  gradually  lose  vitality 
if  those  who  use  them  separate  themselves  from  realities. 
If  there  is  an  enforced  separation  in  childhood,  we  cannot 
wonder  that  this  separation  continues  in  after  life. 

Any  work  which  is  not  suited  to  the  receiving  and  trans- 
forming power  of  the  learner's  mind  is  harmful.  Growth 
is  a  matter  of  time  and  nutrition,  but  if  we  work  with 
nature,  the  child  will  do  easily  what  he  might  otherwise  be 
unable  to  do  at  all. 


ADVANCED     AKITHMETIO. 


-ooi^c 


d   c    b  a 


1.  Tell  all  you  can  about  the  units  a,  b,  c,  and  d. 

2.  What  ratios  do  you  see  ? 

3.  2  is  the  ratio  of  which  units  ? 

4.  Compare  h  with  each  of  the  other  units.    Compare  a 
with  each.     Compare  c  with  each.      Compare  d  with  each. 

5.  If  c?  is  1,  each  of  the  other  units  equals  what  part 
of  1? 

6.  What  is  the  ratio  of  1  to  |  ?  to  i  ?   of  f  to  i  ?  of  1 
to  I  of  i  ?  of  1  to  i  of  f  ? 

7.  What  IS  the  sum  of  i  and  ^  ?   of  ^  and  f  ? 

8.  Make  sentences  like  this  :   1  is  the  ratio  of  -J-  of  1  to 
foff. 

9.  Make  sentences  like  this  :  2  is  the  ratio  of  -J^  to  J  of  f . 
10.    If  the  unit  1  costs  ic/,  what  part  of  a?/  does  each  of 

the  others  cost  ? 


2  ADVANCED   ARITHMETIC. 

11.  If  the  unit  i  costs  7/,  the  cost  of  each  of  the  other 
units  equals  how  many  7^  ? 

12.  If  c  is  1,  what  is  the  name  of  each  of  the  other  units  ? 

13.  Why  do  you  call  a  -J  ?    Why  did  you  before  call  it  ^  ? 

14.  What  is  the  ratio  of  1^  to  -J-  ?    to  |  ? 

15.  What  is  the  ratio  of  1  to  ^  ?   to  ^  of  |  ? 

16.  What  is  the  ratio  of  f  to  ^  ?   to  |  ?   to  IJ  ? 

17.  If  6  is  1,  what  is  each  of  the  other  units  ? 

18.  Compare  each  with  the  other  three. 

19.  Why  do  you  call  cZ  2  ?    a  ^7 

20.  c  equals  how  many  halves  of  ^  ? 

21.  3  is  the  ratio  of  c  to  what  part  oi  b? 

22.  What  is  the  ratio  of  l^-  to  ^  ? 

23.  When  d  is  4,  what  is  a? 

24.  When  d  is  2,  what  isb? 

25.  Cut,  free-hand,  out  of  paper,  the  units  1,  J,  -J-,  and  ^. 

26.  Compare  these  units  until  familiar  with  their  ratios. 

27.  Compare  ^  of  each  unit  with  ^  of  each  of  the  others. 
Write  the  comparison. 

28.  What  is  the  ratio  of  1  to  ^  ?   to  ^  ?   of  f  to  ^  ? 

29.  Look  at  a,  h,  c,  and  d.  What  is  the  ratio  of  a  to  -J 
of  a  ?   of  ^>  to  i  of  ^  ?    of  c  to  ^  of  c  ?   oid  toxoid? 

30.  What  is  the  ratio  of  i  to  ^  of  i  ?   of  |-  to  ^  of  -J  ? 

31.  What  is  the  ratio  of  100  to  ^  of  100  ?  to  ^  of  100  ? 
to  i  of  50  ? 

32.  The  ratio  of  a  smaller  to  a  larger  floor  equals  the 
ratio  of  -J-  of  ^  to  ^.  20  yd.  of  carpet  are  required  to  cover 
the  smaller  floor.  How  many  yards  are  required  to  cover 
the  larger  ?  What  is  the  ratio  of  the  time  required  to 
paint  the  larger  floor  to  the  time  required  to  paint  the 
smaller  ? 


ADVANCED   ARITHMETIC. 


t    n    s 


t  s 


1.  In  each  group,  if  n  is  1,  what  is  the  name  of  each  of 
the  others  ? 

2.  In  each  group  touch  the  -J-,  the  2,  the  |,  the  1. 

3.  Observe  a  group  and  think  the  order  in  which  the 
units  are  placed.     Observe  another  ;  another. 

4.  Observe  each  group  and  tell  the  names  from  right 
to  left ;  from  left  to  right. 

5.  Practice  thinking  the  units  from  right  to  left  and 
from  left  to  right  in  each  group. 

6.  Try  to  think  the    order  in  which    each   group    is 
arranged  without  observing  the  units. 

7.  Call  s  1  and  name  the  other  units  in  each  group. 
Touch  the  i,  I,  1,  f. 

8.  Practice  imaging  the  units  in  each  group  in  order. 

9.  Call  jf  1  and  name  the  units  in  each  group. 

10.  Practice  imaging  the  units  in  order. 

11.  Take  a  set  of  solids  which  you  can  call  \,  ^,  f,  1. 
Arrange  in  a  group.  Tell  the  names  from  right  to  left,  or 
left  to  right,  without  observing.  Arrange  again  and  think 
or  tell  names. 

12.  How  many  different  arrangements  of  the  units  in 
a  group  of  four  solids  can  you  make  ? 

13.  If  1  of  a  bunch  of  firecrackers  is  worth  2/,  what  is 
f  of  a  bunch  worth  ? 

14.  f  of  a  pound  of  candy  is  worth  30/.  What  is  ^  of 
a  pound  worth  ? 


ADVANCED   ARITHMETIC. 


I  is  the  relation  of 

J  is  the  relation  of 

2-30/ is  the  cost  of 


1.  What  is  the  ratio  of  |  to  |  ? 

2.  What  is  the  ratio  of  f  to  |  ? 

3.  What  is  the  ratio  of  -|-  to  J  ? 

4.  What  is  the  ratio  of  f  to  |  ? 

5.  What  is  the  ratio  of  -J  to  ^  ? 

6.  2  is  the  relation  of  what  to  ^  ? 
what  to  ^  ? 

7.  f  is  the  relation  of  what  to  f  ? 
what  to  f  ? 

8.  30/  is  the  cost  of  ^  lb.  of  tea. 
how  much  tea  ? 

9.  What  is  the  sum  of  $i  and  $i  ?  of  ^  lb.  and  i  lb.  ? 
of  J-  oz.  and  -J-  oz.  ? 

10.  -J-  and  what  equal  f  ? 

11.  I  and  what  equal  |  ?  li  ? 

12.  i  and  what  equal  f  ? 

13.  The  ratio  of  the  amount  of  work  done  by  A  to  that 
done  by  B  equals  the  ratio  of  f  to  f .  What  is  the  ratio  of 
the  money  A  earns  to  the  money  B  earns  ?  of  -J  of  the 
money  B  earns  to  -J-  of  the  money  A  earns  ? 

14.  What  is  the  ratio  of  the  cost  of  2  oranges  to  the 
cost  of  3  oranges  ?  of  the  cost  of  ^  of  a  dozen  to  the  cost 
of  f  of  a  dozen? 


ADVANCED  AEITHMETIC. 


Ill         IL 


d     c     b     a 


1.  What  ratios  do  you  see?  Write  the  ratios  that 
you  see. 

2.  If  c  is  1,  what  is  the  name  of  each  of  the  other 
units  ? 

3.  Why  do  you  call  b  ^?  a^?  d^? 

4.  If  d  is  1,  what  is  the  name  of  each  of  the  other  units  ? 

5.  Compare  the  1  with  each  of  the  others.  Compare 
the  f  with  each.  Compare  the  ^  with  each.  Compare 
the  i  with  each. 

6.  What  is  the  ratio  of  ^  to  f  of  f  ?  to  ^  ?  to  i  ? 

7.  i  of  -J-  equals  what  part  of  1  ?  ^  of  ^  equals  what 
part  of  1? 

8.  Show  me  the  blackboard.  Show  |-  of  it.  Show  ^  of 
it.     Show  f  of  it. 

9.  Compare  the  blackboard  with  its  f,  its  ^,  its  i. 
Look  at  the  unit  |. 

10.  What  is  the  name  of  the  two  equal  units  in  the 
unit  I  ?  What  four  equal  units  do  you  see  in  the  unit  J  ? 
What  unequal  units  do  you  see  ?  The  units  f  and  ^  are 
unequal.  ^ 

IL  Compare  corresponding  parts  of  the  units.  Ux.  ^ 
of  one  with  -J  of  another. 

12.  The  sum  of  ^  and  ^  of  the  cost  of  a  horse  equals 
what  part  of  its  cost  ? 

13.  i  yd.  of  ribbon  costs  20/.  f  of  20/  equal  the  cost 
of  what  part  of  1  yd.  ? 


6  ADVANCED    ARITHMETIC. 

14.  B  and  A  own  an  equal  amount  of  land.  A  sells  | 
of  his  and  B  i,  at  the  same  price  per  acre.  What  is  the 
ratio  of  the  money  B  receives  to  the  money  A  receives  ? 


1.  What  ratios  do  you  see  ? 

2.  If  the  largest  unit  is  1,  what  is  the  name  of  each  of 
the  others  ? 

3.  What  is  the  ratio  of  1  to  each  of  the  others  ?  of  f  to 
each  ?  of  ^  to  each  ?  of  i  to  each  ? 

4.  What  part  of  ^  do  you  see  in  the  ^  ?     What  part  of 
f  do  you  see  in  the  ^  ?  in  the  J  ? 

5.  What  is  the  sum  of  ^  and  |-  ?  of  f  and  i?  of  f  and 
i?  ofi  +  i  +  i?     How  many  i^'s  in  f  ? 

Ans.   There  are  |  of  ^  in  f . 

6.  How  many  f  in  1  ?     ^  is  how  much  less  than  f  ? 

7.  Compare  100  with  f  of  100 ;  with  ^  of  100 ;  with 


125. 


8.  What  is  the  ratio  of  ic  to  J  of  y?  to  f  ?  to  f  ? 

9.  Make  three  questions  similar  to  the  following  :  If  I 
paid  $i  for  some  ribbon,  and  $f  for  a  book,  what  did  they 
both  cost  ? 

10.  Make  three  questions  like  the  following :  If  1^  lb. 
of  coffee  cost  xf^,  what  part  of  ic/  does  1  lb.  cost  ? 

11.  One  ton  of  coal  weighs  how  many  times  as  much  as 
i  of  a  ton?  It  weighs  how  many  times  as  much  as  -J-  of 
•J-  of  a  ton  ?  It  weighs  how  many  times  as  much  as  -J  of  f 
of  a  ton  ? 

12.  Four  is  the  ratio  of  1  ton  of  coal  to  what  ?  Four  is 
the  ratio  of  1  ton  of  coal  to  what  part  of  ^  ton?  Four 
is  the  ratio  of  1  ton  of  coal  to  what  part  of  f  ? 


ADVANCED   ARITHMETIC. 


4    "-"   -    ^ 

7.   If  the  ratio  of  |  to  |  is  m,  what  is  the  ratio  of  J  of 


1.  What  is  the  ratio  of  |  to  |  ? 

2.  What  is  the  ratio  of  f  to  f  ? 

3.  What  is  the  ratio  of  J  to  |  ? 

4.  What  is  the  ratio  of  |  to  f  ? 

5.  What  is  tlie  ratio  of  ^  to  |-  ? 

6.  If  the  ratio  of  |  to  f  is  a,  what  is  the  ratio  of  2 •}  to 
3?  of  2'#  to2-3? 

Itoioff? 

8.  The  ratio  of  the  cost  of  1  yd.  of  silk  to  the  cost  of 
1  yd.  of  muslin  is  3.  What  is  the  ratio  of  the  cost  of  20 
yd.  of  silk  to  the  cost  of  20  yd.  of  muslin  ? 

9.  If  at  the  above  rates  a  lady  pays  an  equal  amount 
for  muslin  and  for  silk,  what  is  the  ratio  of  the  number  of 
yards  of  muslin  purchased  to  the  number  of  yards  of  silk? 

10.  The  ratio  of  the  rectangle  m  to  the  rectangle  h  is  3. 
What  is  the  ratio  of  the  sum  of  the  two  rectangles  to  5  ? 
to  m  ?  What  is  the  greatest  part  of  m  which  is  an  exact 
measure  of  each  ? 

11.  The  ratio  of  a  rectangle,  a,  to  the  rectangle  h  is  1. 
What  is  the  ratio  of  the  sum  of  -J  of  S  and  f  of  a  to  a  ? 
to  6? 

12.  X  and  ^  are  two  equal  rectangles.  The  greatest 
common  measure  of  |-  of  ic  and  of  f  of  ?/  is  what  part  of  y  ? 
what  part  of  f  of  3/  ? 

13.  A  merchant  cuts  two  pieces  of  lace,  one  8  yd.  in 
length,  the  other  6  yd.,  into  the  longest  equal  remnants. 
Each  remnant  equals  what  part  of  6  yd.  ?  what  part  of  8 
yd.  ?     What  is  the  ratio  of  ^  of  6  yd.  to  J  of  8  yd.  ? 


8 


ADVANCED    ARITHMETIC. 


14.  What  is  the  ratio  of  i  of  1  yd.  to  f  of  1  yd.  ?  The 
greatest  common  measure  of  ^  of  1  yd.  and  i  of  1  yd. 
equals  what  part  of  -J  of  1  yd.  ? 

15.  Draw  equal  rectangles  of  different  dimensions  and 
show  the  ratio  of  ^  of  1  to  -J  of  the  other. 

16.  The  floor  a  equals  f  of  f  of  the  floor  b.  What  is 
the  ratio  of  the  amount  of  work  done  in  sweeping  the  two 
floors  ? 


1.    Draw  a  rectangle  and  separate  it  into  eight  equal 
parts.     Show  the  two  largest  equal  parts  of  the  rectangle. 
Show    the    four    largest   equal    parts. 
Show  I",  J,  f ,  and  }  of  the  rectangle. 

2.  What  is  the  ratio  of  the  rectangle 
to  each  of  these  parts  ?  of  -J  of  it  to 
each  ?  of  f  of  it  to  each  ? 


3. 
4. 
5. 

toi? 
6. 


What  is  the  ratio  of  2i  to  i  ?  to  f  ? 

What  is  the  ratio  of  5  to  2^  ?  to  1 J  ? 

3  is  the  ratio  of  what  to  ^  ?     2  is  the  ratio  of  what 


2  is  the  ratio  of  what  to  f  ? 


What  is  the  ratio  of 


li  to  f  ? 

7.  A  farmer  has  1^  bu.  of  red  clover  seed  and  f  bu. 
of  white.     How  much  more  of  the  red  than  the  white  ? 

8.  If  1^  bu.  of  corn  is  worth  $x,  what  part  of  $ic  is  1 
bu.  worth  ? 

9.  What  is  the  ratio  of  1  bu.  to  1  pk.  ?  of  J  bu.  to 
J  pk.  ?  of  I  bu.  to  I  pk.  ? 

10.  What  is  the  ratio  of  a  2-in.  cube  to  a  cu.  in.  ?  of  i 
of  a  2-in.  cube  to  ^  of  a  cu.  in.  ?  of  ^  of  a  2-in.  cube  to  \ 
of  a  cu.  in.  ? 

11.  Draw  a  2-in.  square.  Draw  a  sq.  in.  What  is  the 
ratio  of  the  2-in.  square  to  the  sq.  in.?  of  J  of  the  2-in. 
square  to  J  of  the  sq.  in.  ? 


ADVANCED    ARITHMETIC. 


Wa 


1.    What  ratios  do 


Write  the  ratios  that 


you  see 
you  see. 

2.    If  c  is  1,  what  is  the  name  of  each  of  the  others  ? 


3.  What  is  the   ratio   of   If  to   each  of   the   others  ? 
of  1^  ?    of  1  ?   of  I  ?  of  i  ? 

4.  If  h  is  1,  what  is  each  of  the  others  ? 

5.  Learn  these  names  :  li,  1,  f ,  \,  \. 

6.  Compare  1  with  each  of  the  others.     Compare  \ 
with  each. 

7.  If  a  is  1,  what  is  each  of  the  others  ? 

8.  What  is  the  ratio  of  f  to  each  of  the  others  ?  of  f  ? 
of  1? 

9.  If  the  f  equal  45  sq.  ft.,  what  part  of  45  sq.  ft.  does 
each  of  the  others  equal  ? 

10.  If  the  f  is  worth  $a?,  what  is  each  of  the  other  units 
worth  ? 

11.  What  is  the  ratio  of  f  of  15  to  f  of  15  ?  of  f  of  ic  to 
f  of  iC? 

12.  If  X  is  7  times  as  large  as  y^  \  oi  x  is  how  many 
times  as  large  as  \  of  ?/?  What  is  the  ratio  of  5  a?  to  5  ?/  ? 
of  :i-  of  ic  to  ^  of  ?/  ? 

13.  Use  different  sets  of  prisms  having  the  ratios 
1,  2,  3,  4,  5.  Use  blackboard  drawings  having  the  same 
relations. 


10 


ADVANCED   ARITHMETIC. 


1.  What  is  the  ratio  of  |  to  |  ?  of  f  to  |  ?  of  i  to  J  ? 

2.  What  is  the  ratio  of  |  to  |  ?  of  f  to  |  ?  of  ^  to  J  ? 

3.  What  is  the  ratio  of  |  to  |  ?  of  |  to  f  ?  of  J  to  J  ? 

4.  What  is  the  ratio  of  |  to  |  ?  of  f  to  f  ?  of  J  to  J  ? 

1.  There  are  two  spoons  at  each  plate.  If  one  spoon  is 
taken  from  each  plate,  what  part  of  all  the  spoons  is 
taken  ? 

2.  If  you  give  Mary  -^  of  each  of  the  apples  on  a 
plate,  what  part  of  the  apples  do  you  give  her  ?  How  do 
you  divide  the  apples  into  two  equal  parts  ?  If  you  give 
her  J  of  each  apple,  what  part  of  all  the  apples  do  you 
give  her  ? 

3.  Separate  an  apple  into  two  equal  parts.  Divide  the 
apple  equally  between  3  boys.  If  you  give  each  boy  -J-  of 
each  half,  what  part  of  the  apple  does  he  receive  ?  Each 
half  is  divided  into  how  many  parts  ? 

4.  Separate  a  piece  of  paper  into  two  equal  parts. 
Divide  the  two  parts  equally  between  5  pupils.  If  you 
give  each  pupil  ^  of  each  part,  what  part  of  the  paper  does 
he  receive  ? 

5.  Separate  a  piece  of  paper  into  five  equal  parts.  If 
you  give  Harry  ^  of  each  5th  of  the  unit,  what  part  of  the 
unit  do  you  give  him  ? 

6.  -J  of  each  |  of  5  equals  what  part  of  6? 


ADVANCED    ARITHMETIC. 


11 


ii 


IP' 


1.  What  ratios  do  you  see  ? 

2.  How  often  does  the  ratio  3  occur  ?  the  ratio  2  ? 

3.  Find  the  unit  of  which  d  is  the  largest  exact  measure. 

4.  What  is  the  largest  common  measure  of  e,  c,  and  a  ? 
e  is  the  largest  common  measure  of  which  units  ? 

5.  What  part  of  e  is  the  largest  common  measure  of 
e  and  d  ? 

6.  What  part  of  d  is  the  largest  measure  of  d  and  c  ? 

7.  If  we  call  a  1,  what  is  the  name  of  each  of  the  other 
units  ? 

8.  Show  the  different  units  and  tell  the  names. 
What  is  the  ratio  of  1  to  each  of  the  other  units  ? 
What  is  the  ratio  of  each  to  1  ?  of  |-  of  each  to  -J- 


9. 

10. 

of  1? 

11. 

12. 


What  is  the  ratio  of  1  bu.  to  1  pk.?  of  i  bu.  to  ^  pk.? 
What  is  the  ratio  of  1  sq.  yd.  to  1  sq.  ft.  ?  of  ^^ 
sq.  yd.  to  J  sq.  ft.  ? 

13.  What  is  the  ratio  of  1  yd.  to  1  ft.  ?  of  ^  yd.  to  i  ft.  ? 

14.  What  is  the  ratio  of  any  part  of  1  yd.  to  a  corre- 
sponding part  of  1  ft.  ? 

15.  If  the  ratio  of  the  rectangle  b  to  the  rectangle  a  is  x, 
what  is  the  ratio  of  any  part  of  the  rectangle  a  to  a  corre- 
sponding part  of  the  rectangle  b  ? 

16.  To  paint  a  blackboard  equal  to  the  rectangle  a  costs 
$6  ;  to  paint  one  equal  to  the  rectangle  b  costs  $5.  What 
is  the  ratio  oi  b  to  a? 


12  ADVANCED   ARITHMETIC. 

17.  What  is  the  ratio  of  the  time  required  to  paint  |-  of 
h  to  the  time  required  to  paint  ^  of  a  ? 

18.  Observe  the  above  units.     If  a  is  1,  what  is  Z*  ? 

19.  What  is  the  ratio  of  |  to  each  ?  of  each  to  |  ? 

20.  What  is  the  ratio  of  |  to  each  of  the  others  ?  of  ^  ? 
of  J? 

21.  Draw  lines  having  the  same  ratios  as  the  solids  above. 
Show  the  line  whose  ratio  to  d  equals  the  ratio  of  e  to  <Ly 
of  c  to  S  ;  of  /  to  a. 

22.  The  line  a  can  be  separated  into  how  many  parts, 
each  equal  to  /?  the  line  c  ? 

23.  The  sum  of  what  two  lines  eqitals  the  longest  line  ? 

24.  Write  sentences  similar  to  this  :  the  sum  of  \  and  ^ 
equals  ^. 

25.  Make  sentences  like  this  :  -J  equals  |  of  \. 

26.  Look  at  the  J.     What  part  of  J  do  you  find  in  \  ? 

27.  What  part  of  |  in  -J  ?  what  part  of  the  |  in  ^  ? 

28.  What  part  of  |  in  each  unit  ? 

29.  What  is  the  ratio  of  6  to  1  ?  of  1  to  J  ?  of  1  to  6  ? 
of  J  to  1  ? 

30.  What  is  the  ratio  of  2  to  3  ?  of  ^  to  ^  ?  of  3  to  2  ? 
of  itoi? 

31.  What  part  of  J  is  the  largest  common  measure  of  J 
and  ^  ?  How  many  of  these  measures  in  -J-  ?  in  ^  ?  What 
part  of  1  is  the  largest  common  measure  of  ^  and  ^  ? 

32.  What  is  i^  of  i  ?  What  is  i  of  i  ?  |  of  ^  equals 
what  ?  ^  of  f  equals  what  ?  What,  then,  is  true  of  f  of 
^  and  i  of  I  ? 

33.  f  of  ^  of  1  yard  is  used  for  a  croquet  ground,  and  ^ 
of  f  of  it  is  a  flower  bed.  What  is  the  relative  size  of  the 
flower  bed  and  of  the  croquet  ground  ? 

1.  What  is  the  ratio  of  1  to  |  ?  to  |  ?  to  f  ? 

2.  What  is  the  ratio  of  1^  to  f  ?  to  ^  ?  to  f  ? 


ADVANCED    ARITHMETIC.  13 

3.  What  is  the  ratio  of  1  to  IJ  ?  to  1^?  to  IJ?  to  1|  ? 

4.  What  is  the  ratio  of  ^  of  ic  to  J  of  o:^  ? 

5.  What  is  the  ratio  of  |  of  a?  to  ^  of  a?  ? 

6.  What  is  the  ratio  of  f  of  7  to  ^  of  7  ? 

7.  What  is  the  ratio  of  |  of  19^  to  ^  of  19^  ? 

8.  What  is  the  ratio  of  7  to  f  of  7  ?  of  9  to  f  of  9  ? 

9.  What  is  the  ratio  of  ic  to  f  ? 

10.  What  is  the  ratio  of  J  to  ^  of  J  ?  of  i  to  J  of  i  ? 

11.  What  is  the  ratio  of  i  to  f  of  i  ?  of  J  to  f  of  J  ? 

12.  Make  problems  similar  to  the  following  :  If  a  slate 
costs  $  J  and  a  book  $  J,  what  is  the  cost  of  both  ? 

13.  How  many  strips  of  carpet  ■}  yd.  long  can  be  cut 
from  a  strip  6  yd.  long  ? 

14.  At  s;/  for  ^  doz.  oranges,  J  doz.  costs  what  part  of 
z^  ?  What  is  the  ratio  of  the  cost  of  i  doz.  to  the  cost  of 
f  doz.  ? 

15.  X  equals  the  cost  of  ^  barrel  of  flour.  What  part  of 
X  equals  the  cost  of  f  of  a  barrel  ? 

16.  Mrs.  Jones  paid  f  of  ^  of  her  money  for  a  dress  and 
^  of  f  of  it  for  a  cloak.  What  was  the  ratio  of  the  cost  of 
the  dress  to  the  cost  of  the  cloak  ?  What  part  of  her 
money  did  she  spend  ? 

17.  At  36/  a  dozen,  what  is  the  cost  of  ^  of  J  of  a  dozen 
oranges  ?  What  is  the  cost  of  one  orange  ?  What  is  the 
ratio  of  ^^  of  a  unit  to  -^  of  ^  of  the  unit  ? 

18.  The  ratio  of  a  larger  unit  to  a  smaller  is  70.  What 
is  the  ratio  of  J  of  the  larger  to  J  of  the  smaller  ? 

19.  What  is  the  ratio  of  a  rectangle  3  by  2  to  a  rect- 
angle 2  by  2  ?  What  is  the  ratio  of  ^  of  the  larger  rect- 
angle to  ^  of  the  smaller  ?  If  an  amount  equal  to  J  of  the 
larger  rectangle  were  added  to  the  smaller  rectangle,  what 
would  then  be  the  ratios  of  the  rectangles? 

20.  What  is  the  ratio  of  8  oz.  to  1  lb.  ?  of  ^  of  8  oz.  to 
ilb.? 


14 


ADVANCED    ARITHMETIC. 


21.  What  is  the  ratio  of  f  of  a  cord  of  wood  to  |  of  a 
cord  ?  of  -J  cord  to  ^  cord  ? 

22.  A  man  bought  a  farm  for  %x  and  sold  f  of  it  for  an 
amount  equal  to  what  he  paid  for  it.  What  was  the  ratio 
of  the  amount  he  received  for  \  of  the  farm  to  the  amount 
he  paid  for  the  farm  ?  of  the  amount  he  received  for  \  of 
the  farm  to  the  amount  he  paid  for  |-  of  it  ? 


1.  Draw  these  rectangles  on  the  blackboard,  making  e 
1  sq.  ft. 

2.  Write  all  the  ratios  that  you  see. 

3.  If  a  is  1,  what  is  each  of  the  other  units  ? 

4.  Compare  1  with  each  of  the  others.    Compare  \  with 
each.     Compare  f  with  each.     Compare  \  with  each. 

5.  If  c  is  1,  what  is  the  name  of  each  of  the  others  ? 

6.  Learn  the  names  1^,  f ,  1,  |-,  and  \.     Compare  each 
with  the  other  four. 


1.  What  is  the  largest  equal  part  of  the  unit  that  you 
see  ?  JSTame  all  the  equal  parts  that 
you  see. 

2.    Compare  the  unit  with  each  of 
the  parts.     Compare  |  with  each  of 
the  other  parts.      Compare  f  with 
each.     Compare  \  with  each.     Com- 
pare \  with  each. 

3.  \  equals  what  part  of  -J  ?  of  |  ? 

4.  What  part  of  |  do  you  find  in  J  ?  \\\  \1  in  |  ? 


B 

B 

B 

p 

w 

J 

ADVANCED   ARITHMETIC. 


15 


5.  What  is  the  ratio  of  1  to  |  ?  of  1  to  1 1  ? 

6.  Show  me  |  of  the  unit ;  the  | ;  the  J. 

7.  Look  at  the  rectangle 
What  three  equal  units  ? 


What  two  equal  units  in  |  ? 


1.  What  is  the  ratio  of  |  to  |  ?  of  J  to  }  ?  of  J  to  i  ? 

2.  What  is  the  ratio  of  f  to  f  ?  of  |  to  |  ?  of  ^  to  J? 

3.  What  is  the  ratio  of  | 


to  f  ?  of  f  to  I  ?  of  ^  to  1  ? 

4.  What  is  the  ratio  of  f 
to  I  ?  of  f  to  f  ?  of  5  to  I  ? 
of  4  to  f  ?  of  f  to  3  ?  of  f  to 
I  ?  of  1  to  1  ? 

5.  X  is  the  ratio  of  |  of  the 
distance  to  |-  of  the  distance  traveled  in  an  hour.     What 
is  the  ratio  of  \  of  the  distance  to  \  of  the  distance  ? 

6.  f  is  the  ratio  of  the  weight  of  a  package  of  sugar  to 
a  package  of  coffee.  What  is  the  ratio  of  two  packages, 
one  weighing  \  as  much  as  the  sugar  and  the  other  \  as 
much  as  the  coffee  ? 

7.  f  of  the  unit  a  equals  the  unit  h.  \  of  the  unit  a 
equals  what  part  of  the  unit  h  ?  What  part  of  a  is  an 
exact  measure  of  both  a  and  h  ?  what  part  of  ^  ? 

8.  The  sum  of  ^^  of  a  and  \  oih  equals  what  part  of  a? 
of  ^? 

9.  If  \  of  the  width  of  a  room  equals  \  its  length, 
what  is  the  ratio  of  the  length  to  the  width  ? 

10.  Make  an4  answer  similar  problems  :  The  ratio  of  two 
rectangles  is  f .  What  part  of  the  smaller  is  an  exact  meas- 
ure of  both  ?  of  the  larger  ? 

11.  The  ratio  of  2  to  3  equals  the  ratio  of  \  to  what  ? 

12.  The  ratio  of  6  to  5  equals  the  ratio  of  \  to  what  ? 

"  Advance  in  representativeness  of  thought  makes  possible 
advance  in  abstractness."  —  Herbert  Spencer. 


16 


ADVANCED   ARITHMETIC. 


1.  Select  solids  that  represent  1,  |,  f,  ^,  J,  J  and  study 
until  the  relations  of  these  units  can  be  given  readily. 

2.  Select  other  sets  having  the  same  ratios  and  master 
the  relations. 

3.  Make  similar  sentences  :  If  J  weighs  13  lb.,  ^  weighs 
I  of  13  lb.  If  the  cost  of  f  of  1  bu.  of  potatoes  is  80^,  the 
cost  of  i  bu.  is  f  of  80/. 

1.  Draw  a  rectangle  and  separate  it  into  two  equal 
parts. 

2.  Separate  each  half  into  thirds.  J  of  -^  equals  what 
part  of  a  rectangle  ? 

3.  Separate  each  6th  into  halves.    ^  of  J  equals  what  ? 

4.  Show  the  two  largest  equal  parts  of  the  rectangle. 
Show  the  three  largest  equal  parts.  Show  the  six  largest 
equal  parts.     Show  J,  J,  i,  f  j  I?  and  the  1. 

1.  Draw  a  rectangle  and  separate  it  into  10  equal  parts. 
Two  of  these  parts  equal  what  part 
of  the  rectangle  ? 

2.  Show  me  the  |  of  a  rectangle. 
Show  the  1 ;  the  f  ;  the  | ;  the  f  ; 
the  1. 

3.  Compare  each  with  the  other  four. 

4.  Compare  1  with  IJ  ;  with  | ;  with  f  ;  with  f . 

5.  What  is  the  ratio  of  J  to  ^  of  ^  ?  of  ^  to  ^  of  |  ? 

6.  What  is  the  ratio  of  5  to  1  ?  of  1  to  J  ? 


ADVANCED   ARITHMETIC.  17 

7.  The  largest  common  measure  of  |  and  ^  of  |  equals 
what  part  of  the  rectangle  ? 

8.  Make  sentences  similar  to  the  following  :  The  sum 
of  f  and  j\  equals  /_  of  the  rectangle.  J  of  |  of  the 
rectangle  equals  f  of  ^  of  the  rectangle. 

9.  What  is  the  ratio  of  f  to  j%  ?  of  f  to  j%  ?  of  J 
to^V? 

10.  What  is  the  ratio  of  |  to  f  ?  of  |^  to  i  ? 

11.  What  is  the  ratio  of  7  to  ^^  ?  of  i  to  ^\  ?  of  a  to 
5?  of  Stoiof§? 

1.  If  6  doz.  peaches  cost  $f,  how  many  dozen  can  be 
bought  for  $1  ? 

2 
40  _  Head :  Four  thirds  of  6  equals  the  number 

$  '      of  dozen  that  can  be  bought  for  $1. 

2.  Make  and  express  many  comparisons,  thus :  If  9 
cost  40/,  f  of  9  can  be  bought  for  $1.  Why  ?  If  4  can 
be  bought  for  $^,  |  of  4  can  be  bought  for  $1.  If  in  f  of 
a  box  there  are  9  doz.,  there  are  f  of  9  doz.  in  a  box. 

3.  What  is  the  ratio  of  $1  to  $J  ?  What,  then,  is  the 
ratio  of  $4  to  $f  ?  j  is  the  ratio  of  $1  to  what  ?  4' }  is 
the  ratio  of  $4  to  what  ? 

4.  If  the  cost  of  3  pk.  of  potatoes  is  $f ,  how  many 
pecks  can  be  bought  for  $4  ? 

2  What  is  the  ratio  of  $1  to  $f  ?     What, 

4  5   3  _  then,  equals  the  number  that  can  be  bought 

>       "  for  $1  ?  for  $4  ? 

What  is  the  ratio  of  30  pk.  to  3  pk.  ?  of  $4  to  $f  ? 

What  is  the  ratio  of  3  pk.  to  30  pk  ?  of  $f  to  $4  ? 

5.  If  30  pk.  of  potatoes  cost  $4,  how  many  pecks  can 
be  bought  for  $f  ?  What  part  of  30  pk.  can  be  bought  for 
$1  ?    What  part,  then,  of  i  of  30  pk.  can  be  bought  for  $f  ? 

6.  At  $f  each,  how  many  cakes  can  be  bought  for  $6  ? 


18  ADVANCED    ARITHMETIC. 

7.  At  $f  a  yard,  how  many  yards  of  merino  can  be 
bought  for  $8? 

8.  At  $1  a  bushel,  how  many  bushels  of  potatoes  can 
be  bought  for  $6  ?  for  $7  ? 

9.  A  boy  earns  $|  in  1  day.  In  how  many  days  does 
he  earn  $8  ? 

10.  A  man  mows  f  of  an  acre  of  grass  in  1  hour.  In 
how  many  hours  does  he  mow  5  acres  ? 

11.  At  $f  a  yard,  how  many  yards  of  silk  can  be  bought 
for  $3  ? 

12.  At  12^/  each,  how  many  collars  can  be  bought 
for  $3  ? 

Practice  thinking  and  writing  the  ratios  in  the  follow- 
ing until  you  can  do  both  without  hesitation.  Write 
the  ratios,  nothing  more.  Ex.  Write  | ;  not  |  is  the  ratio 
of  1  yd.  to  2  ft. 

1.  What  is  the  ratio  of  1  yd.  to  2  ft.  ?  to  1^  ft.  ?  to 

4  ft.?  to  IJ  yd.?  to  2i  yd.?  to  7  ft.?  to  1  yd.?  to  2  ft.? 

2.  What  is  the  ratio  of  1  bu.  to  3  pk.  ?  to  4  pk.  ?  to  5 
pk.  ?  to  li  bu.  ?  to  IJ  bu.  ?  to  17  pk.  ?  to  2^  bu.  ? 

3.  What  is  the  ratio  of  1  sq.  yd.  to  9  sq.  ft.  ?  to  7  sq. 
ft.  ?  to  6  sq.  ft.  ?  to  10  sq.  ft.?  to  IJ  sq.  ft.  ? 

4.  What  is  the  ratio  of  a  nickel  to  3/?  to  4/?  to  7/? 
to  9/  ?  to  10^  ? 

5.  What  is  the  ratio  of  1  lb.  to  8  oz.  ?  to  12  oz.  ?  to  14 
oz.  ?  to  4  oz.  ?  to  20  oz.  ?  to  6  oz.  ?  to  f  lb.  ?  to  f  lb.  ?  to 
f  lb.  ?  to  I  lb.  ?  to  li  lb.  ?  to  17  lb.  ? 

6.  What  is  the  ratio  of  1  cu.  yd.  to  27  cu.  ft.  ?  to  18 
cu.  ft.  ?  to  24  cu.  ft.  ?  to  1|  cu.  ft.  ? 

7.  What  is  the  ratio  of  a  2-in.  sq.  to  3  sq.  in.  ?  to 

5  sq.  in.  ?  to  7  sq.  in.  ? 

8.  What  is  the  ratio  of  1  yd.  to  f  yd.  ?  to  f  yd.  ?  to  f 
yd.  ?  to  f  yd.  ?  to  1^  yd.  ?  to  3f  yd.  ? 


ADVANCED    ARITHMETIC.  19 

9.    What  is  the  ratio  of  $1.  to  $i  ?  to  $f  ?  to  $2^  ?  to 

10.  What  is  the  ratio  of  $1  to  $1.37^?   to  $.37^?    to 
$.62i?  to$.87i?  to$.12i?  to$1.37i? 

11.  What  is  the  ratio  of  $1  to  16|^?   to  66^^^?   to 
831^? 

12.  What  is  the  ratio  of  $1  to  20/  ?  to  60/  ?  to  80/  ? 

13.  What  is  the  ratio  of  1  to  f  ?  to  f  ?   to  J  ?  to  1^  ? 
to  I  ?  to  li  ?  to  f  ?  to  I  ?  to  li  ?  to  f  ?  to  li  ? 

14.  What  is  the  ratio  of  1  to  }?  to  J?  to  J  ?  to  f?  to 
V?  to  3J?  to  17i?  to  18i?  to  23f  ? 

15.  What  is  the  ratio  of  1  to  |  ?  to  f  ?   to  §  ?   to  f  ? 
to  f  ?  to  I  ?  to  t\  ?  to  If  ?  to  if  ?  to  -V^-  ?  to  "-^^  ? 

16.  What  is  the  ratio  of  1  to  .3  ?  to  .7  ?  to  .5  ?  to  .2  ? 
to  .8  ?  to  1.2  ?  to  1.5  ?  to  7.5  ?  to  2.3  ? 

1.  If  the  cost  of  2  yd.  of  cloth  is  $J,  how  many  yards 
can  be  bought  for  $5  ? 

5  •  4  ■  2  _  What  is  the  ratio  of  $1  to  $  J  ?    What, 

3      ~      ^"       then,  equals  the  amount  of  cloth  that  can 
be  bought  for  $1  ?  for  $5  ? 

2.  What  is  the  ratio  of  $5  to  $f  ?  of  13J  yd.  to  2  yd.? 

3.  What  is  the  ratio  of  $f  to  $5  ?  of  2  yd.  to  13 J  yd.? 

4.  If  131  yd.  of  cloth  cost  $5,  how  many  yards  can 
be  bought  for  $f  ? 

2  What  is  the  ratio  of  $1  to  $5?     What, 

?-— ^  =  2      then,  equals  the  amount  of  cloth  that  can  be 
^■^■$        *     bought  for  $1? 

5.  The  cost  of  2  yd.  of  cloth  is  $f .     What  is  the  cost 
ofl3iyd.? 

6.  If  13J  yd.  cost  $5,  what  is  the  cost  of  2  yd.  ? 
^•3$  _  What  is  the  ratio  of  1  yd.  to  13  J  yd.  ?   What, 

M      ~       then,  equals  the  cost  of  1  yd.  ?  of  2  yd.  ? 


20  ADVANCED    ARITHMETIC. 

1.  Make  similar  sentences  : 

If  7  can  be  bought  for  12^ f^,  3  '  8  '  7's  can  be  bought  for 
$3.     Why  ? 

What  is  the  ratio  of  $1  to  12^/  ? 

What,  then,  equals  the  number  that  can  be  bought  for 
$1?  for  $3? 

2.  Make  sentences  like  this  : 

If  in  I  of  a  bolt  there  are  10  yd.,  — =  the  number 

of  yards  in  5  bolts. 

3.  Make  sentences  like  thi's  :  ^ 

If  3  boxes  of  oranges  can  be  bought  for  $3^  ($i)j  — i^ 

equals  the  number  of  boxes  that  can  be  bought  for  $14. 

4.  Pupils  question  thus  :  If  3  doz.  plates  can  be  bought 
for  $4^,  how  many  dozen  can  be  bought  for  $15  ? 

Other  pupils  write  : 

5.  Make  sentences  like  this  :  8  *  |  is  the  ratio  of  8  bu. 
to  §  of  a  bushel. 

6.  What  is  the  ratio  of  9  to  f  ?    What  is  the  ratio  of  1 
to  3  ?     What,  then,  is  the  ratio  of  9  to  f  ? 

7.  Practice  writing  the  ratios  of   the  following  until 
you  can  do  so  readily.     Ux.    What  is  the  ratio  of  9  to  f  ? 

Ans.    — . 

8.  What  is  the  ratio  of  2  to  f  ?  of  5  ?  of  9  ?  of  12  ? 

9.  What  is  the  ratio  of  $7  to  $1  ?  of  $2  ?  of  $12  ? 
of  $17  ? 

10.  What  is  the  ratio  of  3  bu.  to  |  bu.  ?  of  7  bu.  ? 
of  11  bu.  ? 

1  If  pupils  cannot  do  this  work  with  reasonable  rapidity,  give  work 
which  they  can  do,  and  approach  this  through  growing  power.  (See 
note,  p.  73,  '' Elementary  Arithmetic") 


ADVANCED    ARITHMETIC.  21 

11.  What  is  the  ratio  of  5  ft.  to  |  ft.?  of  12  ft.?  of 
13  ft.  ? 

12.  What  is  the  ratio  of  2  to  |f  ?  of  5  ?  of  29  ?  of  74  ? 

13.  What  is  the  ratio  of  6  to  y-  ?  of  26  ?  of  39  ?  of  ^  ? 

14.  What  is  the  ratio  of  9  to  -y-  ?  of  5  ?  of  11  ?  of  1 J  ? 

15.  What  is  the  ratio  of  17  to  j%  ?  of  38  ?  of  7  ?  of  2  J  ? 

16.  What  is  the  ratio  of  18  to  j%  ?  of  16  ?  of  144  ? 
off? 

17.  Dictate  problems  thus :   What  is  the  ratio  of  100 

Pupils  write  : —  =  the  ratio  of  100  to  ^2^. 

1.  If  there  are  x  qt.  in  f  pk.,  how  maiiy  x  qt.  in  8  pk.  ? 

2.  If  there  are  6  qt.  in  f  pk.,  how  many  qt.  in  12  pk.  ? 

=  the  number  of  qt.  in  12  pk. 

o 

By  what  comparisons  and  inferences  was  the  above  state- 
ment obtained  ? 

3.  There  are  10  in.  in  |  ft.    How  many  inches  are  there 
in  12i  ft.  ? 

4.  There  are  14  oz.  in  f  lb.     In  9  lb.  how  many  ounces  ? 

5.  There  are  6  sq.  ft.  in  |  of  1  sq.  yd.      ^^ — - —  equals 

the  number  of  square  feet  in  how  many  square  yards  ? 

6.  There  are  21  cu.  ft.  in  J  cu.  yd.    How  many  f  of  21 
cu.  ft.  in  15  cu.  yd.  ? 

7.  There  are  12  cu.  ft.  in  |  cu.  yd.     How  many  cubic 
feet  in  17  cu.  yd.  ? 

8.  If  there  are  48  cu.  ft.  in  f  cd.,  how  many  cubic  feet 
in  7  cd.  ? 

Remark.  —  Let  pupils  write  similar  problems.  Call  on  other 
pupils  to  give  statements,  indicating  their  solution  as  in  prob- 
lem 2. 


22  ADVANCED   ARITHMETIC. 

1.  What  is  the  ratio  of  |  to  f  ?   of  J  to  i  ?   of  f  to  f  ? 
of  1  to  i  ?  of  I  to  f  ?   of  i  to  1  ? 

2.  Cut  two  rectangles  whose  ratio  equals  the  ratio  of  | 
to  f  ;  of  i  to  J. 

3.  The  ratio  of  |  to  |  equals  the  ratio  of  what  part  of 
jtoj? 

4.  The  ratio  of  12  a  to  a  equals  the  ratio  of  -|^  to  what 
part  of  a? 

Practice  expressing  the  ratio  of  a  to  b,  and  of  ^  to  a  at 

3  '5  2 ' 1  . 

sight.     Ux.   —^  is  the  ratio  of  a  toh  \  ^-^^  is  the  ratio  of 

h  to  a. 

a  h  a  h 

1.  3  f  6  f 

2.  7  I  7  I 

3.  4  I  16  i 

4.  10  ^^^  13  f 

5.  4  I  f  3 

1.  If  X  is  the  amount  that  can  be  bought  for  $6,  what 
is  the  amount  that  can  be  bought  for  $f  ? 

%' X  _  X  What  equals  the  amount  that  can  be  bought 

r0  ~  2l'    for  $1  ?  for  $f  ? 
3 

2.  What  is  the  ratio  of  $f  to  $7  ? 

2 '  1 

— -  =  the  ratio  of  $f  to  $7. 

o    I 

What  is  the  ratio  of  $1  to  $7  ?    What,  then,  is  the  ratio 

of  $f  to  $7  ? 

What  is  the  ratio  of  — 

3.  j  to  4  ?  7.  21  to  11  ?  11.  2   to  21  ? 

4.  f  to  7  ?  8.     .7  to    8  ?  12.  ^^  to  15  ? 

5.  I  to  5  ?  9.     .3  to    5  ?  13.  |  to  14  ?      . 

6.  f  to6?         10.  1.2  to  13?  14.  I  tol7J? 


ADVANCED    ARITHMETIC.  23 


What  is  the  ratio  of  3  sq.  ft.  to  IJ  sq.  ft. 


? 


3  '  3  _  ^  What  is  the  ratio  of  1  sq.  ft.  to  Ij  sq.  ft.  ? 

4    ~"    **      What,  then,  is  the  ratio  of  3  sq.  ft.  to  IJ 
sq.  ft.  ? 

What  is  the  ratio  of  — 

2.  7  sq.  ft.  to  2i  sq.  ft.  ?      12  sq.  ft.  to  4?  sq.  ft.  ? 

3.  6  sq.  ft.  to  31  sq.  ft.  ?        .7  sq.  ft.  to   .7  sq.  ft.  ? 

4.  9  sq.  ft.  to  2f  sq.  ft.  ?      3f  sq.  ft.  to  7^  sq.  ft.  ? 

5.  12  sq.  ft.  to  3|  sq.  ft.  ?    13i  sq.  ft.  to  7J  sq.  ft.  ? 

6.  2 J  sq.  ft.  to  21  sq.  ft.  ?    47^  sq.  ft.  to  24f  sq.  ft.  ? 
What  is  the  ratio  of  — 

7.  8  cu.  ft.  to  2i  cu.  ft.  ?       12  cu.  ft.  to  17  cu.  ft.  ? 

8.  12  cu.  ft.  to  2f  cu.  ft.  ?      .5  cu.  ft.  to  14^  cu.  ft.  ? 

9.  2i  cu.  ft.  to  7i  cu.  ft.  ?     f  cu.  ft.  to  14^  cu.  ft.  ?  . 

10.  What  is  the  ratio  of  a  2-in.  sq.  to  f  sq.  in.  ? 
44 

11.  What  is  the  ratio  of  a  3-in.  sq.  to  |  sq.  in.  ? 

12.  What  is  the  ratio  of  a  4-in.  sq.  to  f  sq.  in.  ? 

13.  What  is  the  ratio  of  a  6-in.  sq.  to  |  sq.  in.  ? 

14.  What  is  the  ratio  of  a  4-in.  sq.  to  f  sq.  in.  ? 

15.  What  is  the  ratio  of  a  6-in.  sq.  to  f  sq.  in.  ? 

16.  What  is  the  ratio  of  1  sq.  ft.  to  f  of  a  6-in.  sq.  ? 

17.  What  is  the  ratio  of  a  2-in.  cube  to  f  cu.  in.  ? 

18.  What  is  the  ratio  of  a  3-in.  cube  to  f  cu.  in.  ? 

When  graduates  of  high  schools  and  normal  schools  assert  that 
the  distance  around  a  rectangle  containing  3  rows  of  7  sq.  ft.  is 
21  sq.  ft. ;  that  a  box  required  to  hold  a  set  of  3-in.  cubes  should 
be  3  times  the  size  of  a  box  required  to  hold  a  similar  set  of  1-in. 
cubes  ;  and  when  these  instances  are  not  exceptional  but  typical, 
does  it  not  seem  that  contact  with  realities  in  their  mathematical 
aspect  is  a  necessity  ? 

When  it  is  equally  common  for  pupils  who  can  tell  that  the 
ratio  of  a  to  b  is  x  to  be  unable  to  tell  that  the  ratio  of  5  a  to  5  6 


24  ADVANCED    ARITHMETIC. 

is  X,  or  i  of  a  to  1  of  ^  is  x,  is  it  not  fair  to  infer  that  the  pres- 
entation is  restricting  rather  than  promoting  mental  action,  that 
it  is  preventing  seeing  things  as  they  are  ?  ^ 

For  a  lesson  write  five  problems  and  questions  on  each, 
similar  to  the  following  : 

1.  If  the  cost  of  2  doz.  roses  is  $f ,  how  many  can  be 
bought  for  $14? 

2 

X^ '  8  '  2  _  What  equals  the  number  of  dozen  that 

'^       ~  can  be  bought  for  $1  ?  for  $14  ? 

2.  If  the  cost  of  6   qt.  of  berries   is   $f,  how   many 
quarts  can  be  bought  for  $4  ? 

t 

(a)  The  cost  of  36  qt.  is  $4.  How  many  quarts  can  be 
bought  for  $1? 

(b)  The  cost  of  36  qt.  is  $4.    What  is  the  cost  of  6  qt.  ? 

(c)  The  cost  of  6  qt.  is  $f .    What  is  the  cost  of  36  qt.  ? 

Remark. —  If  you  observe  the  second  problem,  you  will  see  that 
it  suggests  the  questions  (a),  (h),  and  (c).  Ask  similar  questions 
on  each  of  the  following  : 

3.  If  the  cost  of  5  yd.  of  cloth  is  $f ,  how  many  yards 
can  be  bought  for  $7  ? 

4.  If  a  5-ft.  staff  casts  a  shadow  f  of  a  rod  long,  what 
is  the  length  of  the  shadow  cast  by  a  staff  7  ft.  high  ? 

5.  If  the  cost  of  3  lb.  of  coffee  is  $|,  how  many  pounds 
can  be  bought  for  $4  ? 

1  "  Whenever  I  went  far  enough,  I  touched  geometrical  bottom."  — 
Professor  Sylvester. 

"  The  higher  processes  of  mind  in  mathematics  lie  at  the  very 
foimdation  of  the  subject."  —  Professor  Sylvester. 


ADVANCED    ARITHMETIC.  25 

6.  If  X  bu.  of  wheat  are  required  to  make  3  bbl.  of 
flour,  how  many  bushels  are  required  to  make  4  bbl.  of 
flour  ? 

7.  At  the  rate  of  69  mi.  in  3^  hr.,  how  far  does  a  car 
run  in  8  hr.  ? 

8.  X  equals  the  number  of  miles  a  car  runs  in  5^  hr. 

7  ■  3  ■  X' 
In  what  time  does  the  distance  it  runs  equal  — — —  ? 

9.  Write  three  problems.  Ask  three  questions  on 
each.     (See  problem  2.) 

1.  If  15  yd.  of  calico  can  be  bought  for  $f ,  how  many 
yards  can  be  bought  for  $|  ? 

2.  What  is  the  ratio  of  1  to  f  ? 

The  ratio  of  f  to  f  equals  what  part  of  the  ratio  of  1  to 
f  ?     What  is  the  ratio  of  f  to  f  ? 

3.  What  is  the  ratio  of  }  to  |  ? 

4.  Make  similar  sentences  :  |  of  |,  or  ||,  is  the  ratio  of 

T   ^^    9- 

Write  the  ratios  of  the  following,  thus : 

4  8        32      ^-         4.-      4.  .  .    . 
^TT^j  or  —  =  the  ratio  of  |  to  |. 
o   7         oo 

What  is  the  ratio  of  — 

5.  /^to  I?  9.     I  to-y?         13.     l-tof? 

6.  \^  to  J?  10.     i  to  Y-  ?         14.    IJ  to  f  ? 

7.  HtoJ^V?         11.     I  to  8?  15.   If  to  ^9^? 

8.  f  to  f  ?  12.    j\  to  7  ?  16.    it  to  9  ? 

1.   What  equals  the  ratio  of  2|  to  3^  ? 


2 

5  ■  ^      ^'     equals  the  ratio  of  2  J,  or  y^,  to  3^  ? 


J'^  •  2  _  ^  What  is  the  ratio  of  1  to  3^  ?     What,  then, 


26  ADVANCED   ARITHMETIC. 

2.  If  X  yd.  of  cloth  can  be  bought  for  $3J,  how  many 
yards  can  be  bought  for  $2|  ? 

2 

X4^'2'x  _^x 

What  equals  the  ratio  of  — 

3.  2ito6i?         7.    5ito2i?  11.     6§  to    9  j  ? 

4.  7f  to9J?         8.   3fto7i?  12.  33i  to  51§  ? 

5.  41  to  9  J  ?        9.   24  to  31  ?  13.  13f  to  17}  ? 

6.  2f  to  3§  ?       10.   5J  to  7f  ?  14.  13f  to  5  ? 

1.  If  the  cost  of  10  yd.  of  lace  is  $16f,  how  many 
yards  can  be  bought  for  $52f  ? 

Suggestion.  —  After  finding  the  answer  to  this  question,  make 
and  solve  three  other  problems  suggested  by  the  question. 

2.  If  a  man  earns  $2f  in  1  day,  in  how  many  days 
does  he  earn  $13|  ? 

3.  If  I  yd.  of  silk  is  enough  for  3  neckties,  8j  yd.  are 
enough  for  how  many  neckties  ? 

4.  If  a  man  pays  $f  a  day  for  his  meals,  in  how  many 
days  does  he  spend  $16|-  ? 

5.  If  a  family  uses  |  of  a  barrel  of  flour  in  1  wk.,  how 
long  do  5J  barrels  last  ? 

6.  If  the  cost  of  X  lb.  of  coffee  is  $7,  how  many  pounds 
can  be  bought  for  $7f  ? 

7.  At  $J  a  pound,  how  many  pounds  of  coffee  can  be 
bought  for  $2  ?  for  $5  ?  for  66f  /  ? 

8.  At  $f  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $18  ? 

9.  At  $5^  a  cord,  how  many  cords  of  wood  can  be 
bought  for  $84  ? 

10.    How  many  pieces  of  cloth  |  yd.  long  can  be  cut 
from  a  piece  10  yd.  long  ? 


ADVANCED    ARITHMETIC.  27 

11.  How  many  yards  of  ribbon  at  $i  each  can  be  bought 
for  $J  ? 

12.  .1  is  the  ratio  of  what  to  .7  ?  to  2.4  ? 

13.  The  ratio  of  .3  to  .5  equals  the  ratio  of  3  to  what  ? 

14.  The  ratio  of  j  to  |  equals  the  ratio  of  what  to  J  ? 

15.  Alice  has  a  piece  of  green  ribbon  9  yd.  in  length 
and  a  piece  of  blue  6  yd.  in  length.  She  wishes  to  cut 
them  into  the  longest  equal  pieces  possible.  When  cut, 
what  is  the  ratio  of  each  piece  to  9  yd.  ?  of  each  piece  to 
6  yd.  ? 

1.  What  is  the  ratio  of  2^  to  .7  ? 
5 

5'X(I>  _  25  _  What  is  the  ratio  of  1  to  .7  ?    What, 

^  •  7  ~"  7  ~    ^'      then,  is  the  ratio  of  2^,  or  f,  to  .7  ? 

What  is  the  ratio  of  — 

2.  9ito.3?  9    to    .5? 

3.  3J  to  .8  ?  2f  to  1.7  ? 

4.  18J  to'  .7  ?  I  to    .9  ? 

5.  ejtol.l?       13^  to  2.5? 

6.  7|tol.3?         5fto4.3? 

7.  What  is  the  ratio  of  1.2  (|f )  to  9^  ? 

12-  ^  _  12  ^         What  is  the  ratio  of  1  to  91-  ?     What, 
af0  •  19  "  95  *      then,  is  the  ratio  of  1.2  to  9^-  ?     j%  is  the 
5  ratio  of  1  to  what  ?     1.2  of  y2_  is  the  ratio 

of  1.2  to  what  ? 

What  is  the^ratio  of  — 

8.  .3  to    9i?        .5  to    9? 

9.  .8  to    31  ?     1.7  to    22  ? 

10.  .7  to  18i  ?        .9  to      I  ? 

11.  1.1  to    6|  ?     2.5  to  13}  ? 

12.  1.3  to    7f?     4.3  to    5J? 


28 


ADVANCED    ARITHMETIC. 


1.  If  a  is  1,  each  of  the  other  units  equals  what  part 
of  1  ?  Compare  each  unit  with  every  other  unit  in  the 
diagram. 

2.  What  is  the  ratio  of  J  to  |-  of  ^  ?  of  J  to  J  of  ^  ?  of 
itof  ofi?  of  |tof  of  |?'of|toiof|?  offtoiof  §? 

3.  What  is  the  ratio  of  i  to  J  of  i  ? 

4.  What  is  the  ratio  of  1  to  J  of  J  ?  of  J  of  J  to  1  ? 

5.  What  is  the  ratio  of  -J  to  ^  of  -J-  ? 
What,  then,  is  the  ratio  of  1  to  ^  of  -J-  ?  oi  ^  oi  ^ 


6. 
tol? 

7. 


What  is  the  ratio  of  9  to  ^  of  ^  of  9 


"  A  great  part  of  the  progress  of  formal  human  thought  .  .  . 
has  been  due  to  the  invention  of  what  we  may  call  short-mind 
symbols.  .  .  .  But  it  should  never  be  forgotten  that  the  mighty 
stenophrenic  engine  of  which  we  here  speak,  like  all  machinery, 
affords  us  rather  a  mastery  over  nature  than  an  insight  into  it ; 
and  for  some,  unfortunately,  the  higher  symbols  of  mathematics 
are  merely  brambles  that  hide  the  living  springs  of  reality. 

"Many  of  the  greatest  discoveries  of  science — ^^for  example, 
those  of  Galileo,  Huygens,  and  Newton  —  were  made  without  the 
mechanism  which  afterwards  becomes  so  indispensable  for  their 
development  and  application."  —  T.  J.  McCormack,  Open  Court, 
December,  1897. 


ADVANCED   ARITHMETIC.  29 

8.  What  is  the  ratio  of  2  to  i  of  |  of  2  ? 

9.  What  is  the  ratio  of  1  to  ^  of  ^  of  1  ? 
10.    What  is  the  ratio  of  a?  to  -J  of  -J  of  x  ? 

1.  What  is  i  of  i  of  1  ?  What  is  the  ratio  of  J  to  |- 
of  J  ?  What,  then,  is  the  ratio  of  1  to  J  of  ^  ?  What, 
then,  is  the  ratio  of  J^  of  J  to  1  ? 

2.  What  part  of  ic  is  J  of  J  of  cc  ? 

3.  What  is  the  ratio  of  a:  to  J  of  i  of  a?  ? 

4.  What  part  of  an  apple  is  J  of  ^  of  it  ? 

5.  What  is  the  ratio  of  a  unit  to  -^  of  ^  of  the  unit  ? 

6.  What  part  of  1  ft.  is  J  of  ^  of  a  ft.  ?     Why  ? 

7.  What  is  the  ratio  of  2  ft.  to  J  of  i  of  2  ft.  ? 

8.  What  is  the  ratio  of  cc  to  J  of  i  of  a?  ? 

9.  What  part  of  a  unit  is  J  of  ^  of  the  unit  ?  i  of  i  ? 
lofi?  lof  1?  ^VofxV?  .lof.l?  iof  J^? 

10.    If  J  of  1  is  ^^  of  a  unit,  what  part  of  a  unit  is  i 
off? 

"  All  the  materials  of  intellect  are  images  and  symbols,  all  its 
processes  are  operations  on  images  and  symbols."  —  Lewes. 

Operations  on  symbols  are  barren  without  the  experiences  which 

give  them  significance.      To  judge  of  the  ratio  of  f  and  |  these 

symbols  should  call  up  such  distinct  images  of  the  things  that 

their  ratio  is  at  once  grasped.      Until  this  ratio  is  seen  it  cannot 

furnish  a  groundwork  for  conclusions  concerning  things.      To 

know  vaguely  the  meaning  of  |,  of  J,  and  J,  is  not  enough  ;  we 

nmst  see  |  and  ^  in  the  relation  upon  which  our  inference  is 

based.    For  example,  if  a  pupil  is  expected  to  see  that  a  man  who 

5  '  $8 
receives  $8  for  doing  |  of  a  piece  of  work  should  receive  — j — 

for  doing  |  of  it,  he  should  have  had  experiences  which  cause  him* 

to  so  realize  the  relation  of  |  to  §,  that  — r —  will  at  once  suggest 

■*  • 

itself  as  a  necessary  relation. 


30  ADVANCED    ARITHMETIC. 

11.  If  i  of  f  equals  ^^  of  a  unit,  what  part  of  a  unit  is 

|of  f? 

12.  To  what  part  of  a  unit  is  f  of  f  of  it  equal  ? 

13.  What  is  I  of  f  ?     f  of  t-\  equals  what  ? 

14.  What  is  J  of  f  ?     What  is  -J  of  ^  ? 

15.  What  part  of  7  is  y\  of  |  of  7  ? 

16.  What  is  the  ratio  of  1  to  ^^  of  f  of  1  ? 

17.  What  is  the  ratio  of  .9  of  f  of  1  to  1  ?     What  is  .1 
off?     What,  then,  is  .9of  |? 

18.  1.2  of  tV  equals  what  ?     What  is  .1  of  /_?     What, 
then,  is  if  of  ^^  ? 

19.  Pupils  write  problems  similar  to  problem  18,  and 
represent  answers  by  drawings. 


1. 

A  of  1  =  ? 

TlVof 

¥  =  ? 

2. 

A  of  «  =  ? 

i  of 

13^  =? 

3. 

^^  of  ^Sj  =  ? 

i  of 

211  =? 

4. 

tS  of  a  =  ? 

i  of 

281  =? 

5. 

1  of  21  =  ? 

1  of 

421  =  ? 

6. 

J   of  ^T  =  ? 

i   of 

3Ti  =? 

7. 

I§  of  f  f  =  ? 

IS  of 

A  =? 

8. 

Aof¥  =  ? 

f  of 

II   =  ? 

9. 

.9  of  f  f  =  ? 

1.3  of 

f  =? 

.0. 

f  of8§  =  ? 

1     I  of 

15i  =? 

See  note,  p.  12,  "  Elementary  Arithmetic." 

Review  the  work,  pp.  258-260,  "  Elementary  Arithmetic." 

1.  If  X  equals  the  amount  of  work  4  men  can  do  in 
a  day,  5  x  equals  what  ? 

2.  A  can  do  ^^  of  a  piece  of  work  in  1  day  and  B  -J-  of 
it.     What  part  of  the  work  can  both  do  in  a  day  ? 

3.  A  can  mow  x  acres  in  -|-  of  a  day,  and  B  can  mow 
X  acres  in  J  of  a  day.  If  both  work  for  a  day,  what  is 
the  ratio  of  the  number  of  acres  mowed  to  x  acres  ? 


ADVANCED   ARITHMETIC.  31 

4.  John  can  do  a  piece  of  work  in  a  day,  and  James  in 
i  of  a  day.  If  both  work  together  what  is  the  ratio  of  the 
work  they  can  do  in  a  day  to  the  work  done  ? 

5.  If  8  men  mow  7f  acres  of  grass,  how  much  do  3 
men  mow  in  the  same  time  ?  The  ratio  of  3  to  8  equals 
the  ratio  of  what  to  7f  acres  ? 

6.  If  12  men  pave  5-g-  rd.  of  street  in  x  hr.,  how  much 
do  5  men  pave  in  x  hr.? 

4 
5  ■  \^ 

-— -^  =  ?     What  ratios  are  equal  ? 

3 

7.  If  for  $60  you  can  buy  f  of  an  acre  of  land,  for  $12 
you  can  buy  how  much  ? 

8.  If  the  cost  of  f  of  an  acre  of  land  is  $60,  what  is 
the  cost  of  ^^  of  an  acre  ? 

9.  If  $12  is  the  cost  of  ^^  of  an  acre  of  land,  what  is 
the  cost  of  f  of  an  acre  ? 

10.  If  ^\  of  an  acre  of  land  costs  $12,  how  much  can  be 
bought  for  $60  ? 

11.  J  is  the  ratio  of  a  larger  farm  to  a  smaller.  What 
is  the  ratio  of  the  amount  of  money  required  to  buy  J  of 
the  larger  to  the  amount  required  to  buy  J  of  the  smaller  ? 

12.  i^fi-  is  the  ratio  of  the  money  a  man  borrowed  to  the 
interest  he  paid.  What  is  the  ratio  of  the  interest  he  paid 
to  the  money  borrowed  ?  If  he  borrowed  $742,  how  much 
interest  did  he  pay  ? 

13.  $400  equals  ^^  of  the  interest  a  man  pays  in  1 
year.  How  much  interest  does  he  pay  ?  What  is  the 
ratio  of  the  interest  he  pays  to  $400  ? 

14.  A  merchant  sold  goods  for  a  sum  equal  to  J  of  the 
cost.     What  was  the  ratio  of  the  cost  to  the  selling  price? 

15.  What  is  the  cost  of  1 J  yd.  of  cloth  at  $f  a  yard  ? 

16.  At  15^/  a  pound,  what  is  the  cost  of  $4^  lb.  of  beef  ? 


32 


ADVANCED    ARITHMETIC. 


17.  If  a  train  runs  125|  mi.  in  6  hr.,  what  is  the  rate 
per  hour  ? 

18.  In  each  of  the  above  problems  state  what  ratios  are 
equal. 


1.  What  is  the  ratio  of  12  to  6  ?  of  ^  of  12  to  ^  of  6  ? 
of  ^  of  12  to  J  of  6  ?  of  f  of  12  to  f  of  6  ? 
of  f  of  12  to  f  of  6  ? 

2.    What  is  the  ratio  of  56  to  14  ?     What 
is  the  ratio  of  |  of  56  to  i-  of  14  ?     What  is 


6 


true  of  the  ratio  of  56  to  14  and  of  |  of  56 
to  I  of  14  ? 

3.  What  is  the  ratio  of  ^  to  ^  ?     What 
is  the  ratio  of  ^  of  -J-  to  -J^  of  -J- ? 

4.  How   can   the  ratio  of  two  units  be 
found  without  a  direct  comparison  of  the  units  ? 

5.  How  can  the  ratio  of  12  to  15  be  found  indirectly  ? 
What  part  of  each  can  be  found  easily  ?  What  is  -J-  of  12  ? 
of  15  ?     What  is  the  ratio  of  4  to  5  ?     What,  then,  is  the 


ratio  of  12  to  15  ? 


12 
15 


i  of  14  and  ^  of  21  are  corresponding  parts  of  14 


6. 

and  21.     Give  examples  of  corresponding  parts  of  other 
units.     The  ratio  of  j  of  14  to  i  of  21  equals  the  ratio  of 


what  to  21  ? 


What  is  the  ratio  of  i  of  14  to  i  of  21  ? 


What,  then,  is  the  ratio  of  14  to  21  ?  What  ratios  are 
equal  ?  The  ratio  of  the  corresponding  parts  of  two  units 
equals  the  ratio  of  what  ?  The  ratio  of  two  units  equals 
the  ratio  of  what  ? 

7.  What  is  the  ratio  of  J  of  |  to  J  of  f  ?     What,  then, 
is  the  ratio  of  |  to  |  ?     Why  ? 

8.  What  is  the  ratio  of  J  to  ^  ?     What,  then,  is  the 
ratio  of  I  to  I  ?     Why  ? 


ADVANCED    ARITHMETIC.  33 

9.    f  is  the  ratio  of  x  to  y.     What,  then,  is  the  ratio 
of  i  of  ic  to  i  of  ?/  ?    of  i  of  a;  to  ^  of  y  ?     Why  ? 

10.   f  is  the  ratio  of  what   to  45  ?     f  is  the  simplest 


expression  of  the  ratio  of  what  to  45  ?     15 


Express  the  following  ratios  in  the  simplest  form  : 

1.  \%.  5.   If.  9.    -rVs-  13.    if. 

2.  |§.  6.    ||.  10.    H-  14.   m- 

3.  H-  7.   -rV^.  11.    |o.  15.   ij. 

4.  §|.  8.    4f.  12.   f|.  16.   H8- 

•Give  at  sight  the  simplest  form  of  the  following  : 

1.  T^j.  5.   4J.  9.    ^.  13.    §J. 

2.  H-  6.   If.  10.   i|.  14.    fj. 

3.  H-  7.    If.  11.   4|.  15.   IJ. 


4.    if.  8.   iJ.  12.    if  16.   i 

17.  3^5-  19-   4^8-  21.    .15. 

18.  .35.  20.    .625.  22.    .125. 


iL 


1.  Draw  f  and  |.  Draw  -J-  and  ^.  What  is  the  ratio 
of  f  to  I  ?  What,  then,  is  the  ratio  of 
i  to  ^?  Why?  What  parts  are  the 
corresponding  parts  ?  What  is  the  ratio 
of  I  to  f  ?  What,  then,  is  the  ratio  of  J 
toi?  __  _ 

2.  Draw   f   and   f .      Draw  i  and  -J-.      |    2     i    i 
What    is   the   ratio   of   the  f  to  the  f  ? 

What,  then,  is  the  ratio  of  ^  to  :!■  ?  Why  ?  What  is  the 
ratio  of  f  to  |?  What,  then,  is  the  ratio  of  |^  to  ^? 
Why  ?     WhaJ  is  the  ratio  of  ^  to  ^  ?   of  J  to  J  ? 

3.  Draw  J  and  f .  Draw  ^  and  J.  Tell  what  you  can 
about  the  ratios  of  these  units  and  of  their  corresponding 
parts.     What  is  the  ratio  of  |  to  f  ?  of  }  to  |  ?  of  |  to  ^  ? 


34 


ADVANCED    ARITHMETIC. 


of  1  to  1  ?     What  is  the  ratio  of  f  to  |  ?     What,  then,  is 
the  ratio  of  J  to  i  ? 


4.    Draw  units  having  the  relative  size  of  |  and  f  ;  of 
I 


J  and  ^. 


5.    What  is  the  ratio  of  f  to  f  ?   of  ^  to  J  ?   of  f  to  f 


of  1  to  1  ? 

6.  What  is  the  ratio  of  i  to  J  ?    to  J  ?   to  J  ?   to  J  ? 
toi? 

7.  What  is  the  ratio  of  J  to  i  ?   to  1  ?   to  ^  ?   to  i  ? 
toi? 

8.  What  is  the  ratio  of  i  to  i  ?    to  ^  ?    to  i  ?   to  i  ? 

9.  j  is  the  ratio  of  f  to  f .     Why  ? 

10.  Make  many  similar  sentences  :   |  is  the  ratio  of  f 
to  f.     .-.  I  is  the  ratio  of  ^  to  ^. 

11.  What  is  the  ratio  of  f  to  |  ?    of  f  to  |  ?    of  i  to  i  ? 
Why  ? 


ADVANCED   ARITHMETIC.  35 

12.  Make  many  similar  sentences  :  |  is  the  ratio  of  J 
to  1. 

13.  What  is  the  ratio  of  ^%  to  |  ?  of  ^\  to  J  ? 

14.  What  is  the  ratio  of  f  o  to  f  g  ? 

15.  What  is  the  ratio  of  f^  to  Jg  ?  What,  then,  is  the 
ratio  of  ^\  to  J^  ? 

16.  I  is  the  ratio  of  jf  to  |^.     Why  ? 

17.  Make  many  similar  sentences  :  f  is  the  ratio  of  |^ 
to  IS-     •*•  f  is  the  ratio  of  ^\  to  ^\. 

To  Teacher.  —  Train  pupils  to  draw  quickly  units  in  different 
relations.  Through  such  representation  the  condition  of  the 
pupil's  mind  is  revealed.  A  pupil  may  learn  to  sa^/  that  5  is  the 
ratio  of  45  to  9,  or  J  the  ratio  of  ^  to  |^  without  seeing  these 
relations.  One  who  perceives  mathematical  relations  will  be  able 
to  represent  the  magnitudes  compared. 

18.  What  is  the  ratio  of  j%  to  |  ?    of  j\  to  i  ?    of  J  to 

19.  Draw  units  that  represent  the  relative  size  of  J  and  J  ; 
of  j\  and  I ;  of  J  and  J  ;  of  ^\  and  ^^ ;  of  ^\^  and  ^\  ;  of 
I  and  i ;  of  ^\  and  J  ;  of  j\  and  ^\  ;  of  J  and  j\  ;  of  ^\ 
and  J^  ;  of  j\  and  J^. 

20.  Practice  making  mental  pictures  of  units  having  the 
relative  size  of  the  following  :  J  and  ^,  J  and  -J,  ^  and  J, 
^1^  and  j\,  I  and  i,  }  and  J,  ^\  and  |,  ^\  and  y^,  ^^^^  and  .1, 
i  and  .1,  tV  and  .1,  ^^  and  ^V^  ^V  and  ^Vj  A  and  ^V- 

21.  Make  similar  sentences :  The  ratio  of  j^jj  to  ^ 
equals  the  ratio  of  3  to  2. 

1.  What  is  the  ratio  of  5  to  3  ?  of  J  to  ^  ? 

2.  What  is  the  ratio  of  3  to  5  ?  of  ^  to  J  ? 

3.  What  is  the  ratio  of  4  to  5  ?  of  i  to  J  ? 

4.  What  is  the  ratio  of  5  to  4  ?  of  ^  to  |  ? 

5.  What  is  the  ratio  of  4  to  2  ?  of  J  to  ^  ? 

6.  What  is  the  ratio  of  2  to  4  ?  of  ^  to  i  ? 


36 


ADVANCED    ARITHMETIC. 


7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 


What  is  the  ratio  of  6  to  4  ?   of  ^  to  ^  ? 
What  is  the  ratio  of  4  to  6  ?    of  i  to  J  ? 


of  J  to  j\  ? 
of  ^  to  ^^  ? 


What  is  the  ratio  of  9  to  6  ?    of  J  to  i  ? 
What  is  the  ratio  of  6  to  9  ?   of  J  to  i  ? 
What  is  the  ratio  of  18  to  27  ?   of  ^\  to  ^\  ? 
What  is  the  ratio  of  6  to  18  ? 
What  is  the  ratio  of  7  to  28  ? 
What  is  the  ratio  of  12  to  36  ?    of  -^V  to  ^V  '^ 
What  is  the  ratio  of  45  to  9  ?   of  ^V  to  i  -^ 
Make  similar  sentences  :  The  ratio  of  5  to  3  equals 
the  ratio  of  -J  to  J. 

17.  What  is  the  ratio  of  17  to  18  ? 
What  is  the  ratio  of  23  to  37  ? 
What  is  the  ratio  of  ^  to  J  ? 


ofyVtOxV? 


of  A  to  ^V  ^ 


of  J  to 


of  yV  to 


What  is  the  ratio  of  J  to  J  ? 
What  is  the  ratio  of  ^^y  to 


of  ito^V?  of  T-Vtoj? 
1?   of  ^\  to|?   of^^T 


18. 
19. 

20. 
21. 

to  1  ? 

22.  Make  sentences  like  these  :  2  is  the  ratio  of  8  to  4, 
of  i  to  1.  f  is  the  ratio  of  27  to  18,  of  y\  to  ^\.  ^  is  the 
ratio  of  17  to  34,  of  -^^  to  j\. 

23.  Make  sentences  like  these :  The  ratio  of  9  to  12 
equals  the  ratio  of  J^  to  ^.  The  ratio  of  45  to  30  equals 
the  ratio  of  ^^  to  J^. 


Remark.  —  A  unit  equal  to  i  of  8  and  J  of  12  is  the  largest 
exact  measure  common  to  8  and  12.  A  unit  equal  to  ^  of  y^^  or  I 
of  ^  is  the  largest  exact  measure  common  to  the  units  ^^^  and  ^. 
Hereafter  the  term  largest  measure  will  be  used  for  largest  exact 
measure  common. 


ADVANCED    ARITHMETIC. 


37 


1.  f  is  the  ratio  of  a  to  b.      What  part  of  a  is  the 
largest  measure  of  a  and  h  ?    What 
part  of  h  is  the  largest  measure  ? 

2.  If  f  is  the  ratio  of  h  to  k, 
what  part  of  h  is  the  largest 
measure  of  each  ?     Draw  h  and  k. 

3.  If  f  is  the  ratio  of  m  to  n, 

what  part  of  m  is  the  largest  measure  of  each  ? 

4.  f  is  the  ratio  of  -J  to  -J-.  What  part 
of  -J-  is  the  largest  measure  of  each  ? 

5.  Make  many  similar  sentences  :  f  is 
the  ratio  oi  a  to  h.  .*.  J  of  <x  is  the  largest 
measure  of  a  and  h.  What  is  the  ratio  of  J 
of  a  to  I  of  6  ? 

6.  What  is  the  ratio  of  f  to   |  ?    of  ^ 

to  J^  ?     What  part  of  \  is  the  largest  common 

measure  of  each  ?     i  of  y  equals  what  part 

of  1  ? 

7.    What  is  the  ratio  of  f  to  f  ?    of  J  to 
J  ?     What  part  of  J  is  an  exact  measure  of 


Jandof^? 


If  the  pupil  arrives  at  the  fact  that  the  greatest  common  meas- 
ure of  I  and  J  is  y^  by  a  process  which  he  performs  according  to 
order,  a  process  which  does  not  bring  ^  and  ^  into  his  mental  view 
in  such  a  way  that  he  sees  this  to  be  true,  is  not  any  conclusion 
which  is  based  upon  this  purely  formal  for  Mm  ?  He  may  find  and 
say  that  the  sum  of  J  and  ^  =  ^^,  their  difference  Jg,  and  draw 
fact  after  fact,  but  there  is  no  mental  equation,  no  act  which  a 
machine  might  not  perform. 

The  mind  may  operate  freely  and  legitimately  upon  symbols 
when  it  has  first  put  meaning  into  the  symbols.  The  way,  and 
the  only  way,  to  put  meaning  into  symbols  is  by  repeated  acts  of 
sensing,  feeling,  and  thinking.  Through  varied  experiences  ideas 
grow  and  language  becomes  significant. 


38 


ADVANCED    ARITHMETIC. 


8.  What  is  the  ratio  of  ^\  to  I  ?  What  part  of  j\  is 
a  measure  of  each  ?  -J  of  ^^  equals  what  part  of  1  ?  What 
part  of  1  is  a  measure  of  ^^  and  ^  ?  How  many  of  these 
measures  in  ^^^  ?   in  ^  ? 

9.  Draw  the  units  j\  and  i.  What  is  the  part  of  1 
that  is  the  largest  measure  of  ^^  and  J  ?  of  yi^-  and  yi^  ? 
of  j\  and  yV  ?  of  ^  and  1  ?  of  J  and  J  ?  of  J  and  J  ?  of 
yV  and  i  ?    of  J^  and  ^^^  ?   of  yi ^  and  yj^  ?   of  |  and  yV  ? 

10.  If  X  equals  yi^  of  a  and  y  equals  ^  of  a,  what  part  of 
a  is  the  largest  measure  of  x  and  j/  ?  What,  then,  is  the 
part  of  X  that  is  the  largest  measure  of  x  and  y?x  equals 
what  part  of  a  ?  What,  then,  is  the  part  of  a  that  is  the 
largest  measure  ? 

11.  What  is  the  largest  measure  of  ^^y  and  ^  ? 

(a)  it  =  |of||. 

(b)  .•.^iy  =  |of^^. 

(c)  .*.  ^  of  ^ly,  or  J^-,  is  the  1.  m. 

of  Jy  and  yly. 

12.  What  is  the  1.  m.  of  j\  and 
tV?  tV  =  I  of  yi^.  .-.^of  yLis 
the  1.  an.  of  y^^  and  ^\. 


1.   What  is  the  sum  of  |  and  |  ? 


Make  sentences  similar  to  these  :   J  is  the  ratio  of  J  to  J. 
^1^  is  the  1.  m.    |  =  |f .    |  =  ^\.     ||  is  the  sum  of  |  and  f . 
What  is  the  sum  of  — 


2. 

f  and  f  ? 

9. 

fandf? 

16. 

T\and^\? 

3. 

f  andyV? 

10. 

y35and^? 

17. 

1  and  /^  ? 

4. 

|andy\? 

11. 

t\  and  f  ? 

18. 

fandyV? 

5. 

f.andyV? 

12. 

1  and  2  ? 

19. 

1  and  yV  ? 

6. 

land  2^^? 

13. 

^\andi|? 

20. 

T^and/^? 

7. 

f  and  y73  ? 

14. 

f  and  i|? 

21. 

/^  and  1  ? 

8. 

f  and  1  ? 

15. 

1  and  f  ? 

22. 

f  and  yV  ? 

ADVANCED    ARITHMETIC.  39 

23.  What  is  the  sum  of  ^^,  |,  |  ? 

^^     ^^     ^^  =  13  J.       What  is  the  1.  m.  of  |  and  ^  ?   of  ^^^ 

and  yi^^  ?    What,  then,  is  the  1.  m.  of  the  three  units  ? 

I  is  the  ratio  of  J  to  J. 

.-.  ^  of  J,  or  yi^,  is  the  1.  m.  of  the  two  units.     j\  is  a 
measure  of  ^^2- 

.*.  ^^  is  the  1.  m.  of  the  three  units. 

24.  What  is  the  1.  m.  of  |,  |,  ^^,  \\  ? 

Jg  is  the  1.  m.  of  the  9ths,  6ths,  and  18ths. 


.'.  ^^  is  the  1.  m.  of  the  four  units. 

25.    What  is  the  largest  measure  of  f ,  g^,  f ,  ^^,  i|  ? 

J3J-  is  the  1.  m.  of  7ths,  21sts,  14ths,  and  42ds. 

^^  equals  -^%  of  J. 

.'.1  of  4^^,  or  ^1^,  is  the  1.  m.  of  the  five  units. 

2Q.    What  is  the  sum  of  f,  ^2^,  |,  1 1  ? 

Practice   finding  the   1.  m.   in   each  problem.     Practice 
finding  the  sums  in  each : 

1-  H.  A,  I-  8.  f  tV.I,  ii- 

2.  A,  tV.  §•  9-  tV.  tV>  \h  i 

3-  t's,  I,  ^  10.  H.J. /tt,  A- 

4.  J§.  tV.  ^5-  11-  A,  A.  J.  «• 

5.  A,  ii.  i  12.  f,  I,  I,  |. 

6.  f ,  /j,  ^,  ,\.  13.  i,  f,  I,  ■^^,  T^j. 


7.   i,  ^,  I, /r-  14.   I,  I,  A,,', 


Find  the  sum  of  — 

1.  4^  3.   5f  5.   8J 
61  7J                         5? 

2.  3i  4.   6|  6.     4i 
9|  If                      iM 


40  ADVANCED    ARITHMETIC. 

Eind  the  sum  of  — 

1.  4f ,  13|,  and  36|.  6.    35|,  13|,  67i|,  39tV 

2.  35|,  54|,  36f  7.    17^,  27|,  28S,  15|,  37f. 

3.  24f,  48i,27f. 

4.  36i,  59f,  32/2- 

5.  74^83,  18^3^,  60|,  23t§. 

1.  How  much  greater  is  ^f  than  y\  ? 
57  — 28_29     .'.  if  is  f I  greater  than  -/^. 

72  ~  72  '  What  is  the  difference  between  J|  and  -^^  ? 
What  must  be  added  to  ^^  to  make  the  sum  equal  J|  ?  If 
y7^  is  taken  out  of  Jf,  what  is  left  ?  ^f  —  y^^  equals  what  ? 
What  is  the  ratio  of  J|  to  ^^  ?    of  ^7^  to  if  ? 

In  each  what  must  be  added  to  the  lower  unit  to  make 
it  equal  to  the  upper  ? 

2.  H         3.  f         4.  ^7^         5.  A         6.  JJ         7.  if 

13^?  ^T  A 


iV 


8.   What  must  be  added  to  13f  to  make  the  sum  equal 


to  24f  ? 


24f     f  and  what  equal  f  ? 
13|     14  and  what  equal  24  ? 
iOf 

.'.  lOf  must  be  added  to  13f  to  make  the  sum  equal  to 
24f. 

13f  is  how  much  less  than  24f  ?  24f  is  how  much 
greater  than  13f  ?  What  is  the  difference  between  24f 
and  13f  ?  If  13f  is  taken  out  of  24f ,  what  remains  ? 
What  is  the  sum  of  lOf  and  13|  ? 

9.   What  must  be  added  to  24|  to  make  the  sum  equal 
to  73t  ? 

73_9^  4^  and  what  equal  ||  ? 
24|f  25  and  what  equal  73  ? 
48i| 


ADVANCED   ARITHMETIC.  41 

73J  is  how  much  more  than  24|  ?     73J  less  24|  equals 
what  ? 


What  is  the   difference   between  73i  and  24|  ? 
is  the  sum  of  48i  J  and  24§  ? 

10.  23f-|  =  ? 

11.  47i-|  =  ? 

12.  98f  — 15if=? 

13.  1311 -9f  =  ? 

14.  68|- 34519=? 

15.  75^-38H  =  ? 

16.  32ii-18H  =  ? 

17.  68|-35iJ  =  ? 

How  much  greater  is  — 

1.  83|  than  69f  ?  4.    356f  than  178|J  ?  . 

2.  64^  than  35f  ?  5.    125f  than  77|  ? 

3.  28tV  than  13^^  ?  6.    323^  than  84  J  ? 

1.  Draw  a  line  and  separate  it  so  that  the  ratio  of  the 
parts  is  1. 

2.  Fold  a  paper  so  that  the  ratio  of  the  larger  to  the 
smaller  part  is  2.  What  is  the  ratio  of  the  smaller  to 
the  larger  part?  What  is  the  ratio  of  each  part  to  the 
entire  paper  ?    of  the  paper  to  each  part  ? 

3.  What  is  the  ratio  oi  a  to  b?  oibtoa?    — '    . 

a  b 

of  a  to  the  sum  of  a  and  b?    of  ^  to  the  sum  ? 

of  the  sum  to  ^  ?   of  the  sum  to  «  ?     Think  these  ratios 

again  and  again. 

4.  Show  parts  of  the  blackboard  that  have  the  ratio  2. 
What  is  the  ratio  of  each  of  these  parts  to  the  entire  board  ? 
of  the  board  to  each  part  ? 

5.  Observe  a  and  b.     Tell  all  the  . ^ 

^  b  a 

ratios  that  you  see.  The  unit  has  been 
separated  so  that  the  ratio  of  the  parts  is  what  ?  Give  the 
ratio  of  a  and  b-,  of  a  to  the  sum ;  of  &  to  the  sum  ;  of  the 
sum  to  a,  and  of  the  sum  to  b.  What  ratios  have  you 
given  ?  Do  not  observe,  but  image  the  units  and  think 
all  the  ratios  again. 


42 


ADVANCED    ARITHMETIC. 


6.  Show  me  the  circumference  of  the  circle.  Show  me 
the  arc  h ;  the  arc  a.  What  is  the 
ratio  of  arc  a  to  arc  h?  of  b  to  a? 
of  b  to  the  circumference  ?  of  <x  to 
the  circumference  ?  of  the  circum- 
ference to  each  arc  ? 

7.   Draw  a  rectangle.      Separate 
the  rectangle  into  two  parts  so  that 
the  ratio  of  the  rectangle  to  the  larger  part  is  J. 

8.    What  are  the  ratios  of  the  parts  of  this  line  ?   What 
other  ratios  do   you  see?      Think 
these  6  ratios  again  and  again.  -      -•  - 

9.  Draw  a  staff  and  separate  it 
so  that  the  staff  equals  |  of  the  shorter  part.  What  is  the 
ratio  of  the  staff  to  the  longer  part  ?  If  you  break  a  stick 
so  that  the  shorter  part  equals  f  of 
the  longer,  the  longer  equals  what 
part  of  the  entire  stick? 

10.  The  arc  a  is  2  and  the  arc  b 
is  3.  What  ratios  do  you  find  ? 
What  is  the  ratio  of  each  arc  to  the 
circumference  ?  of  the  circumference 
to  each  arc  ? 

11.  Measure  each  part  of  this  line       1 -j — - 

by  -J  of  a.     Discover  all  the  ratios 

that  you  can.    Think  the  ratios  without  observing  the  lines. 

12.  If  a  farm  is  divided  between  a  son  and  a  daughter 
so  that  the  daughter's  portion  equals  J  of  the  son's,  what 
part  of  the  farm  does  each  receive  ? 

13.  Draw  a  line  12  in.  long.  Divide  it  so  that  one  part 
shall  equal  -J  of  the  other.  What  is  the  length  of  each 
part? 

14.  A  and  B  mow  12  acres  of  grass.  A  mows  -J  as  much 
as  B.     How  much  does  each  mow  '^ 


ADVANCED    ARITHMETIC.  43 

15.  A  liat  and  cloak  cost  $12.  The  cost  of  the  hat 
equals  -J  of  the  cost  of  the  coat.  What  is  the  cost  of 
each  ? 

16.  When  the  time  past  noon  equals  -J  of  the  time  to 
midnight,  what  is  the  time  ?  Make  one  drawing  which 
will  illustrate  each  of  the  last  /owr  problems. 


1.  Divide  the  blackboard  so  that  ^  of  one  part  equals 
•J  of  the  other. 

2.  Divide  the  blackboard  so  that  ^  of  one  part  is  as 
large  as  f  of  the  other.     What  ratios  do  you  see  ? 

3.  If  ^  of  the  sugar  in  a  equals  -J  of  the  sugar  in  h, 
what  is  the  ratio  of  the  sugar  in  a  to  the  sugar  in  ^  ? 
What  is  the  ratio  of  the  sugar  in  each  to  the  sugar  in 
both  ?   of  the  sugar  in  both  to  the  sugar  in  each  ? 

4.  If  f  of  the  distance  you  walk  in  the  forenoon  equals 
f  of  the  distance  you  walk  in  the  afternoon,  which  distance 
is  the  greater,  the  forenoon  or  the  afternoon  distance? 
Draw  a  line  to  represent  f  of  the  forenoon  distance.  Draw 
a  line  to  represent  f  of  the  afternoon  distance.  Complete 
each  distance.  Which  is  the  greater  ?  What  is  true  of 
the  length  of  these  lines  ? 

5.  If  f  of  a  equals  f  of  b,  which  is  the  larger  unit  ? 

6.  If  f  of  a  equals  f  of  b,  which  is  the  greater  ? 

7.  If  J  of  ic  equals  \  of  y,  which  is  the  greater  ? 

8.  If  f  of  A's  money  equal  f  of  B's,  and  together  they 
have  $40,  how  much  has  each  ? 

9.  Divide  a  line  30  in.  long  into  two  parts  so  that  their 
relative  length  shall  be  the  same  as  that  of  -J-  of  an  inch 
and  f  of  an  inch. 

10.  Divide  the  unit  48  into  two  parts  so  that  their  rela- 
tive size  shall  be  the  same  as  that  of  §  and  |. 

11.  f  of  a  equals  f  of  b.     Give  ratio  of  b  to  a. 


44  .ADVANCED    ARITHMETIC. 

1.  Draw  the  face  of  the  clock. 

2.  How  many  hours  from  noon  until  midnight  ? 

3.  When  it  is  1  p.m.  what  is  the  time  past  noon  ? 
What  is  the  time  to  midnight  ? 

4.  When  it  is  1  p.m.  the  time  past  noon  equals  what 
part  of  the  time  to  midnight  ? 

5.  When  it  is  1  p.m.  what  is  the  ratio  of  the  time  from 
noon  until  midnight  to  the  time  past  noon  ? 

6.  When  it  is  1  p.m.  what  is  the  ratio  of  the  time  from 
noon  until  midnight  to  the  time  to  midnight  ? 

7.  When  the  time  past  noon  equals  ^j  of  the  time  to 
midnight,  what  is  the  ratio  of  the  time  to  midnight  to  12 
hr.? 

8.  When  it  is  2  p.m.  the  time  past  noon  equals  what 
part  of  the  time  to  midnight  ? 

9.  When  it  is  2  p.m.  what  is  the  ratio  of  12  hr.  to  the 
time  to  midnight  ? 

10.  What  is  the  time  when  the  time  to  midnight  equals 
^  of  the  time  past  noon  ? 

11.  It  is  3  P.M.     Tell  all  the  ratios  that  you  can. 

12.  What  is  the  time  when  the  time  to  midnight  equals 
3  times  the  time  past  noon  ? 

13.  Suppose  it  is  4  p.m.  Tell  the  ratio  of  the  time  past 
noon  to  the  time  to  midnight ;  of  the  time  to  midnight  to 
the  time  past  noon  ;  of  the  time  past  noon  to  the  time  from 
noon  until  midnight ;  of  the  time  from  noon  until  midnight 
to  the  time  past  noon ;  of  the  time  from  noon  until  mid- 
night to  the  time  to  midnight.  Tell  all  these  ratios 
again. 

14.  It  is  5  P.M.     Tell  all  the  ratios  that  you  can. 

15.  What  time  is  it  when  the  time  past  noon  equals  the 
time  to  midnight  ? 

16.  It  is  7  P.M.     Tell  all  the  ratios  that  you  can. 

17.  It  is  10  P.M.     Tell  all  the  ratix)s  you  can. 


ADVANCED    ARITHMETIC.  45 

1.  If  the  area  of  a  circle  equals  the  area  of  a  square, 
and  the  area  of  a  triangle  equals  the  area  of  a  square,  what 
is  the  ratio  of  the  circle  to  the  triangle  ?     Why  ? 

2.  Pupils  show  that  things  which  are  equal  to  the  same 
thing  are  equal  to  each  other. ^ 

If  the  unit  a  equals  f  of  the  unit  b,  and  the  unit  c  equals 
the  unit  h,  what  is  the  ratio  of  «-  to  c  ? 

3.  Give  five  questions  similar  to  the  above. 

4.  What  is  the  relation  of  a^  to  f  of  a?  If  f  of  a 
equals  h,  what  is  the  relation  oi  ato  b?  If  a  equals  J  of  f 
of  a,  why  does  a  equal  |  of  5  ? 

5.  What  is  the  relation  of  cc  to  f  of  ic  ?  If  f  of  a; 
equals  y,  what  is  the  relation  oi  xto  y?  li  x  equals  J  of  f 
of  X,  why  does  x  equal  ^  oi  y? 

6.  What  is  the  relation  of  c  to  f  of  c  ?  If  |  of  c  equals 
d,  what  is  the  relation  oi  do  d?     Why  ? 

7.  What  is  the  relation  of  m  to  ^  oi  m?  If  J  of  m 
equals  o,  what  is  the  relation  of  m  to  o  ?     Why  ? 

8.  What  is  the  relation  of  a  to  1^  a's  ?  If  1^  «-'s  equal 
b,  what  is  the  relation  oi  a  to  b  ? 

9.  Make  and  answer  many  questions  similar  to  the 
following  :  If  1|  a's  equal  b,  what  is  the  ratio  of  a  to  ^  ? 
a  equals  J  of  If  a's  ;  then  a  equals  what  part  of  ^  ? 

10.  If  f  of  a  equals  -J-  of  b,  what  is  the  relation  oi  ato  \ 
oib?     Why  ?     What  is  the  ratio  of  a  to  ^^  ? 

^  I  have  found  many  high  school  pupils  demonstrating  (?)  propo- 
sitions in  geometry  without  any  real  apprehension  of  the  equality 
of  ratios  involved  Vhen  they  assert  that  "  Two  things  each  equal 
to  a  third  are  equal  to  each  other."  Intellectual  haziness  can  be 
cleared  only  by  contact  with  realities.  Such  contact  is  the  basis 
of  inferences  concerning  things  beyond  perception. 

"  Most  of  the  difficulties  in  this  science  are  difficulties  rather 
of  intuition  than  of  reasoning."  —  Lewes,  Problems  of  Life  and 
Mind,  Vol.  I,  p.  388. 


46  ADVANCED    ATIITHMETIC. 

11.  If  f  of  a  equals  f  of  b,  what  is  the  relation  of  a  to  f 
of  6  ?  oi  ato  b?     Show  objectively. 

12.  If  I  of  a  equals  f  of  b,  what  is  the  relation  of  a  to  f 
of  ^  ?  What  is  the  ratio  of  a  to  ^  ? 

13.  If  1^  a's  equal  f  of  b,  what  is  the  relation  of  a  to  f 
of  6? 

To  Teacher.  —  In  solving  the  following  problems  have  pupils 
represent  things  in  given  relations.  Then  review  without  object, 
and  give  similar  problems.  Real  progress  is  shown  by  growing 
power  to  think  of  things  not  present  to  sense. 

1.  If  I  of  A's  capital  equals  f  of  B's,  why  does  the 
ratio  of  A's  capital  to  B's  equal  J  of  f ,  or  J  of  B's  ? 

2.  Two  men  start  from  opposite  points  12  miles  apart 
and  walk  toward  each  other  until  they  meet,  f  of  the  dis- 
tance A  walks  equals  f  of  the  distance  B  walks.  How 
many  miles  does  each  walk  ? 

3.  When  f  of  the  time  past  midnight  equals  f  of  the 
time  to  noon,  what  is  the  hour  of  day  ? 

4.  A  pole,  the  length  of  which  was  120  ft.,  was  in  the 
air  and  water,  f  of  the  length  of  the  part  in  the  water 
equaled  f  of  the  length  of  the  part  in  the  air.  What  was 
the  length  of  the  part  in  the  air  ? 

5.  When  f  of  the  time  past  noon  equals  f  of  the  time 
to  midnight,  what  is  the  hour  of  day  ? 

6.  A  horse  and  carriage  cost  $198,  and  ^  of  the  cost  of 
the  carriage  equaled  f  of  the  cost  of  the  horse.  What  was 
the  cost  of  the  horse  ? 

7.  A  and  B  build  198  rods  of  fence  in  a  certain  time. 
If  -J  of  the  work  A  does  equals  |  of  the  work  B  does,  how 
many  rods  does  each  build  ? 

8.  If  you  know  the  cost  of  a  watch  and  chain  and  the 
part  of  the  cost  of  the  watch  to  which  the  cost  of  the  chain 
is  equal,  how  would  you  find  the  cost  of  each  ? 


ADVANCED    ARITHMETIC.  47 

9.  If  the  cost  of  a  watch  and  chain  were  $x,  and  the 
cost  of  the  chain  were  equal  to  ^  of  the  cost  of  the  watch, 
what  part  of  $x  would  each  cost  ?  If  the  cost  of  the  chain 
were  equal  to  |  of  the  cost  of  the  watch,  what  part  of  $x 
would  each  cost  ? 

10.  A  watch  and  chain  cost  $70,  and  f  of  the  cost  of  the 
watch  equaled  |  of  the  cost  of  the  chain.  What  was  the 
cost  of  each  ? 

11.  John  and  James  saw  a  pile  of  wood  in  x  hours.  If 
James  does  f  as  much  as  John,  what  part  of  the  work  does 
each  do  ? 

12.  A  and  B  can  do  a  piece  of  work  in  10  days.  If 
A  does  f  as  much  as  B,  what  part  of  the  work  does  each 
do? 

13.  How  does  the  entire  amount  of  work  compare  with 
the  part  each  does  in  10  days  ?  How  long  would  it  take 
each  alone  to  do  the  work  ? 

14.  If  A  does  f  as  much  work  as  B,  and  both  together 
earn  $32,  what  is  the  share  of  each  ? 

15.  Two  men  enter  into  a  partnership,  one  investing  If 
times  as  much  as  the  other.  If  they  make  $640,  what  is 
each  man's  share  ? 

16.  If  the  time  past  noon  equals  If  times  the  time  to 
midnight,  the  time  past  noon  equals  what  part  of  12  hours  ? 
What  is  the  hour  of  the  day  ? 

17.  A  stick  was  broken  into  two  pieces  so  that  f  of  the 
longer  piece  equaled  the  shorter;  the  difference  in  the 
length  of  the  two  pieces  was  4  in.  What  was  the  length 
of  the  whole  stick  ? 

18.  A  hat  cost  f  as  much  as  a  cloak,  and  the  difference 
in  their  cost  was  $8.     What  was  the  cost  of  each  ? 

19.  John  and  James  buy  a  sled.  John  pays  f  as  much 
as  James,  and  James  pays  $.12  more  than  John.  How 
much  does  the  sled  cost? 


48  ADVANCED    ARITHMETIC. 

1.  Represent  1  yd.  of  carpet  1  yd.  wide.  Represent  1 
yd.  of  carpet  f  of  a  yard  wide.  What  is  the  ratio  of  the 
former  to  the  latter  ?    of  the  latter  to  the  former  ? 

2.  What  is  the  ratio  of  a  floor  covered  by  20  yd.  of 
carpet  f  of  a  yard  wide  to  a  floor  covered  by  20  yd. 
of  carpet  1  yd.  wide  ? 

3.  If  60  yd.  of  carpet  will  cover  a  floor  equal  to  |  of  a 
floor  to  be  covered,  the  floor  to  be  covered  equals  what  part 
of  the  floor  covered  by  60  yd.  ? 

4.  What  is  the  ratio  of  a  floor  covered  by  x  yd.  of 
carpet  1  yd.  wide  to  a  floor  covered  by  x  yd.  -J  of  a  yard 
wide  ? 

5.  What  is  the  ratio  of  a  floor  covered  by  x  yd.  of 
carpet  1  yd.  wide  to  a  floor  covered  by  x  yd.  \\  yd.  wide  ? 

6.  What  is  the  ratio  of  70  yd.  of  carpet  1  yd.  wide  to 
70  yd.  I  of  a  yard  wide  ? 

7.  If  it  takes  70  yd.  of  carpet  1  yd.  wide  to  cover  a 
floor,  how  many  yards  f  of  a  yard  wide  will  it  take  to  cover 
the  same  floor  ?  70  yd.  f  of  a  yard  wide  will  cover  what 
part  of  the  floor  ?  What  is  the  ratio  of  the  floor  to  f  of  it  ? 
What,  then,  is  the  ratio  of  the  required  number  of  yards 
to  70  yd.  ? 

8.  If  10  yd.  of  carpet  5  ft.  wide  will  cover  a  floor,  how 
many  yards  7  ft.  wide  will  cover  the  floor  ? 

9.  If  a  floor  contains  30  sq.  yd.,  what  part  of  30  yd.  of 
carpet  |  yd.  wide  will  be  required  to  cover  it  ? 

10.  If  a  floor  contains  25  sq.  yd.,  what  part  of  25  yd.  of 
carpet  f  yd.  wide  will  be  required  to  cover  it  ? 

11.  How   many  yards    of   carpet   J   yd.    wide   will   be 
required  to  carpet  a  room  18    ft.  long  and  15  ft.  wide  ? 

12.  How  many  yards  of  carpet  f  yd.  wide  will  carpet  a 
room  24  ft.  long  and  23  ft.  wide  ? 

13.  Write  5  examples  similar  to  the  12th,  and  state  in  one 
sentence  what  equals  the  number  of  yards  required  in  each. 


ADVANCED    ARITHMETIC.  49 

1.  At  $2  a  bushel,  how  many  bushels  of  apples  can  be 
bought  for  $2  ?  At  $4  a  bushel,  what  can  be  bought  for 
$2  ?  Then  what  is  the  ratio  of  the  quantity  of  apples 
which  can  be  bought  for  any  sum  at  $4  a  bushel  to  the 
quantity  which  can  be  bought  for  an  equal  sum  at  $2  a 
bushel  ? 

2.  How  does  the  amount  of  flour  I  can  purchase  for 
$x  when  flour  is  $6  a  barrel  compare  with  the  amount  I 
can  purchase  for  $x  when  flour  is  $12  a  barrel  ?  For  $6 
how  much  flour  can  I  purchase  at  $6  a  barrel?  For 
$6  how  much  flour  can  I  purchase  at  $12  a  barrel  ?  What, 
then,  is  the  ratio  of  the  flour  that  can  be  bought  for  any 
sum  at  $6  a  barrel  to  the  flour  that  can  be  bought  for  an 
equal  sum  at  $12  a  barrel  ?  If  for  $x,  at  $6  a  barrel,  5 
barrels  can  be  bought,  how  much  can  be  bought  for  $x  at 
$12  a  barrel  ? 

3.  What  is  the  ratio  of  the  number  of  cases  of  fruit 
which  can  be  bought  for  $x  at  $15  each  to  the  number  of 
cases  which  can  be  bought  for  $x  at  $5  each  ? 

4.  Write  3  problems  similar  to  the  3d,  and  state  what 
comparisons  you  make  in  solving  them. 

5.  What  is  the  ratio  of  the  flour  which  a  baker  can 
afford  to  put  into  a  6^  loaf  when  flour  is  $3  a  barrel,  to 
the  flour  he  can  put  into  it  when  flour  is  $5  a  barrel  ? 

6.  If  a  10/  loaf  weighs  30  oz.  when  flour  is  $5  a 
barrel,  what  ought  it  to  weigh  when  flour  is  $10  a  barrel  ? 

7.  If  a  5/  loaf  weighs  20  oz.  when  flour  is  $3^,  what 
ought  it  to  wejgh  when  flour  is  $7  a  barrel  ? 

8.  If,  when  flour  is  $6  a  barrel,  you  can  buy  a  28-oz. 
loaf  for  6/,  what  ought  a  6/  loaf  to  weigh  when  flour  is 
$4  a  barrel  ? 

9.  What  is  the  ratio  of  the  number  of  dozen  oranges 
which  can  be  bought  for  $x  at  $f  a  dozen  to  the  number 
that  can  be  bought  for  $ic  at  $f  a  dozen  ? 


50  ADVANCED    ARITHMETIC. 

1.  What  is  the  cost  of  5  pk.  of  gooseberries  at  12-J^/  a 
quart  ? 

2.  At  75^  a  quart,  what  is  the  cost  of  1^  gal.  of 
alcohol  ? 

3.  If  2-J-  lb.  of  butter  cost  50/,  1  lb.  costs  how  many 
5ths  of  50/  ? 

4.  If  5i  bu.  of  beans  cost  $3.30,  what  does  1  bu. 
cost? 

5.  If  John  can  hoe  a  garden  in  1  week,  working  6  hr. 
a  day,  in  what  part  of  a  week  can  he  hoe  it  working  9  hr. 
a  day  ?     Why  ? 

6.  If  12  J  lb.  of  sugar  cost  a-/,  what  do  37^  lb.  cost  ? 

7.  If  7  boys  can  mow  a  tennis  ground  in  10  min.,  how 
long  does  it  take  3  boys  ?     Why  ? 

8.  If  X  yd.  of  carpet  1^  yd.  widejwill  cover  a  floor, 
how  many  yards  f  of  a  yard  wide  will  cover  the  same 
floor? 

9.  Carrie  can  make  a  dress  in  6  days  and  Fannie  in  9 
days ;  if  they  both  work  together  3  days,  what  part  of  the 
work  is  done  ? 

10.  John  can  mow  a  lawn  in  6  hr.  and  Clarence  in  4  hr. 
If  each  work  2  hr.,  what  part  of  the  lot  is  mowed  ? 

11.  Frank,  Charles,  and  Henry  saw  ^  of  a  pile  of  wood 
in  a  day,  and  Frank  and  Charles  saw  ^^^  of  the  pile  in  a 
day.     What  part  of  the  pile  does  Henry  saw  in  a  day  ? 

12.  If  Julia  does  3  times  as  much  as  Jessie,  and  they 
together  do  J  of  a  piece  of  work  in  a  day,  in  what  time 
can  each  do  the  work  ? 

13.  Jane  makes  an  apron  in  f  of  a  day,  and  Lucy  makes 
one  in  J  of  a  day.  How  many  aprons  can  both  make  in  a 
day? 

14.  If  Leroy  can  wash  a  carriage  in  4  hr.,  and  with  the 
help  of  Harry  can  do  it  in  2  hr.,  what  part  of  the  work 
does  each  do  ? 


ADVANCED    ARITHMETIC.  51 

15.  James  and  Kobert  do  a  piece  of  work  in  x  hr.; 
James  does  twice  as  much  as  Eobert.  What  is  the  part 
of  the  work  done  by  each  ?     Kepresent  by  drawing. 

16.  A  man  and  boy  saw  a  pile  of  wood  in  7  hr.,  the 
man  sawing  twice  as  much  as  the  boy.  What  part  of 
the  wood  does  the  boy  saw  ?  In  what  time  could  he 
saw  it  all  ? 

17.  Mr.  Brown  and  his  son  fence  a  lot  in  8  hr.,  the  son 
doing  half  as  much  as  the  father.  In  what  time  could  the 
father  do  the  work  alone  ? 

18.  Draw  a  rectangle  and  separate  it  into  12  equal  parts. 
Think  of  it  as  a  flower  bed  which  Clara,  Olive,  and  Mary 
can  weed  in  2  hr. ;  Clara  and  Olive  in  4  hr. ;  and  Clara  and 
Mary  in  3  hr.  Show  the  part  of  the  flower  bed  Olive  and 
Mary  can  weed  in  1  hr.  In  what  time  can  Olive  and  Mary 
weed  the  bed  ? 

19.  A  can  do  a  piece  of  work  in  4  days  and  B  in  3  days. 
What  is  the  ratio  of  the  work  that  they  can  both  do  in 
4  days  to  the  work  that  A  can  do  in  4  days  ?  What  is  the 
ratio  of  the  work  that  both  can  do  in  3  days  to  the  work 
that  B  can  do  in  3  days  ? 

20.  32  miles  equals  \  of  the  distance  between  two 
places.     32  miles  and  what  equals  \  of  the  distance  ? 

21.  If  %x  equals  the  cost  of  f  of  a  lot,  what  equals  the 
cost  of  f  of  a  lot  ? 

22.  If  t\  of  a  barrel  of  flour  costs  %2\,  what  does  -^  of 
a  barrel  cost  ? 

23.  A  tank  has  two  pipes,  one  filling  it  at  the  rate  of  70 
gal.  an  hour,  the  other  emptying  it  at  the  rate  of  45  gal.  an 
hour.  How  many  gallons  of  water  will  there  be  in  the 
tank  at  the  end  of  6  hr.? 

24.  A  man  walks  18  mi.  while  a  boy  walks  13.  At  this 
rate  how  many  miles  does  the  man  walk  while  the  boy 
walks  65  mi.  ? 


52  ADVANCED   ARITHMETIC. 

25.  A  boy  lost  f  of  his  marbles  and  then  bought  ^  as 
many  as  he  had  left,  when  he  had  12  marbles.  How  many 
had  he  at  first  ? 

26.  The  difference  between  f  and  f  of  the  length  of  a 
line  is  2  in.     What  is  the  length  of  the  line  ? 

27.  The  difference  between  f  and  |  of  a  surface  is  7 
sq.  in.  How  many  square  inches  in  the  area  of  the  sur- 
face ? 

28.  What  unit  is  as  much  greater  than  f  as  f  is  less  ? 

29.  If  to  a  blackboard  you  add  an  amount  equal  to  its  -J-, 
J,  and  30  sq.  ft.  the  sum  equals  three  times  the  number  of 
square  feet  in  the  blackboard,  what  is  its  area  ? 

30.  f  of  a  quart  equals  what  part  of  3  qt.  ?   of  3  gal.  ? 

31.  I  of  a  peck  equals  what  part  of  2  bu.  ?     Keview. 

32.  What  is  the  ratio  of  1  lb.  of  butter  to  4  oz.  ?  to  1  oz.  ? 
to  i  oz.  ? 

33.  -J-  pt.  equals  what  part  of  ^  gal.  ?     Review. 

34.  What  is  the  ratio  of  4  yd.  to  f  of  a  foot  ? 

35.  I  of  1  equals  what  part  of  5  ? 

36.  ^J^  of  1  equals  what  part  of  5  ?   of  4  ?   of  ^  ? 

37.  ^j_  of  1  equals  what  part  of  ^  ?    of  ^  ?   of  J  ? 

38.  6  equals  how  many  4ths  of  8  ? 

39.  5  equals  how  many  4ths  of  7  ? 

40.  6  equals  how  many  hundredths  of  5  ?  of  4  ?  of  -J  ? 
off? 

41.  The  ratio  of  the  number  of  cents  Howard  has  to  the 
number  Lawrence  has  is  f .  The  sum  of  their  money  is 
77^.     How  much  has  each  ? 

42.  Two  boys  bought  a  melon  for  15/.  One  paid  6/ 
and  the  other  9/.  What  part  of  the  melon  should  each 
have  ? 

43.  A  man  has  $2200.  The  ratio  of  the  amount  he  has 
in  gold  to  the  amount  in  notes  is  |.  How  many  dollars  in 
gold  has  he  ?     How  many  in  notes  ? 


ADVANCED    ARITHMETIC.  53 

44.  A  and  B  are  in  partnership ;  their  profits  are  $400. 
A's  investment  equals  If  times  B's.  What  is  each  man's 
share  of  the  profits  ? 

45.  Find  the  cost  of  3000  cedar  posts  at  $2^  per  hun- 
dred. 

46.  What  is  the  rate  per  minute  of  a  train  which  runs 
80  mi.  in  1  hr.  and  40  min.  ? 

47.  If  a  man  earns  |  of  a  dollar  in  §  of  a  day,  how 
much  does  he  earn  in  f  of  a  day  ?  What  ratios  are 
equal  ? 

48.  If  f  of  an  article  is  sold  for  a  sum  equal  to  what  f 
of  it  cost,  what  is  the  ratio  of  the  gain  to  the  cost  ?  Show 
by  drawing. 

49.  A  barrel  of  beef  which  costs  $12.50  is  retailed  at 
8|-/  a  .pound.     The  gain  equals  what  part  of  the  cost  ? 

50.  Out  of  ITGy^^-  tons  of  coal,  how  many  families  can 
be  supplied  with  J  of  a  ton  each  ? 

51.  What  number  diminished  by  |-  of  f  of  itself  leaves 
a  remainder  of  275  ? 

52.  Two  prisms  have  equal  bases.  What  is  the  ratio  of 
the  larger  to  the  smaller,  if  one  is  3|-"  high  and  the  other 
is  2"  high  ? 

53.  Draw  a  square  equal  to  -J-  of  a  square  4"  long.  What 
is  the  ratio  of  the  square  A  to  the  square  B,  if  the  length 
of  A  equals  the  distance  from  the  center  of  B  to  one  of  its 
corners  ? 

54.  How  many  tin  boxes  4"  in  each  dimension  can  be 
put  into  a  box  JL3"  wide,  8"  high,  and  15"  long  ? 

B5.  How  many  prisms  2  by  5  by  7  can  be  put  into  a  box 
4  by  10  by  14  ?   into  a  box  6  by  15  by  21  ? 

56.  How  many  spheres  2"  in  diameter  can  be  put  into  a 
tin  box  6"  by  4"  by  8"  ?  into  a  box  6"  by  4"  by  2"  ?  into 
a  box  6"  by  2"  by  2"  ? 


54 


ADVANCED    ARITHMETIC. 


Place  solids  representing  1,  10, 100,  1000  so  that  the  pupils  can 
handle  and  observe  them.  See  pp.  189-191,  "Elementary  Arith- 
metic," for  method  of  work. 


1.  If  a  is  1,  what  is  o??     If  6  is  1,  what  is  a?     If  c 
is  1,  what  is  Z>? 

2.  If  c  is  1,  what  is  each  of  the  others  ? 

3.  If  b  is  1,  what  is  each,  of  the  others  ? 

4.  If  fHs  1,  what  is  each  ?     If  c?  is  2,  what  is  each  ? 

5.  If  d  is  ^,  what  is  each  ?     If  c?  is  -J,  what  is  each  ? 
If  ^  is  5  ? 

6.  Associate   the   following   names  with  a  variety  of 
units  having  the  ratios  1000,  100,  10,  .1,  .01,  .001 : 


1000 

500 

250 

8000 

2000 

.1    100 

50 

25 

800 

200 

.01    10 

5 

2i 

80 

20 

.001    1 

i 

i 

8 

2 

7.  If  d  is  .5,  what  is  each  of  the  other  units  ? 

8.  If  c  is  4000,  what  is  each  of  the  others  ?  if  it  is 
9000? 

9.  If  h  is  7000,  what  is  each  of  the  other  units  ? 

10.  If  c  is  -1/-,  what  does  each  of  the  other  units  equal  ? 

11.  What  is  the  ratio  of  ^  of  Z>  to  ^  of  each  of  the 
others  ?  of  -^  of  a  ?  of  ^  of  c?  ?  of  :|-  of  c?  to  :^  of  each  of 
the  others  ? 


ADVANCED   ARITHMETIC. 


66 


Place  bundles  of  a  thousand  sticks,  several  bundles  of  hundreds, 
and  tens,  and  a  number  of  ones  where  they  can  be  handled. 


1.  If  we  call  one  of  the  largest  bundles  1,  what  shall 
we  call  the  next  in  size  ?  the  next  ?  What  shall  we  call 
one  of  the  sticks  ? 

2.  Show  a  1.     Show  .1.     Show  .01.     Show  .001. 

3.  Show  .3.     Show  .02.     Show  .004.     Show  .324. 
Give  much  practice  in  each  of  the  following  exercises : 

(a)  Pupils  select  numbers  of  tenths,  hundredths,  and 
thousandths,  and  tell  what  they  have.  JEJx.  Pupil  selects 
.1,  .03,  .002,  and  says,  "1  have  .1,  .03,  .002,  or  132  thou- 
sandths."    Teacher  writes  .132. 

(b)  Write  on  blackboard  and  have  pupils  find  units 
named.     Ex.   Write  .025.     Pupil  finds  .02  and  .005. 

(c)  Pupils  read  the  following  and  find  units  named: 
.02,  .5,  .04,  .040,  .23,  .230,  .1,  .10,  1.2,  .02,  .002,  .2,  .02, 
.020,  .3,  .30,  .300,  .1111. 

(d)  Teacher  or  pupil  dictates  and  pupils  write.  Ux. 
Write  3  tenths  ;  24  tenths  ;  84  tenths  ;  5  hundredths ;  27 
hundredths ;  240  hundredths ;  785  hundredths ;  2  thou- 
sandths ;  456  thousandths ;  400  thousandths ;  2400  thou- 
sandths. 


56  ADVANCED   ARITHMETIC. 

Remark.  —  If,  after  the  above  study,  a  pupil  cannot  write  deci- 
mals, review  the  exercises.  Do  not  explain,  but  keep  their  atten- 
tion on  the  decimal  relations.  The  pupil  will  grow  into  these 
forms  of  expression  as  into  others.  Here,  as  elsewhere,  the 
language  should  be  freely  given,  but  not  made  the  object  of 
thought. 


Tell  the  ratio  of — 

* 

1.  1  to  .1. 

1  to  .01. 

1  to  .001. 

2.  .Ito.Ol. 

.01  to  .1. 

2  to  .2. 

3.  .5  to  5. 

.20  to  .2. 

.3  to  .003. 

4.  .003  to  .3. 

.7  to  .007. 

.001  to  .1. 

5.  .100  to  .1. 

.01  to  .001. 

.04  to  .004, 

1.  How  many  tenths  in  1  ?   in  2  ?   in  1^  ?   in  2^  ? 

2.  How  many  hundredths  in  .1  ?  in  .2  ?  in  .1 J  ?   (Read 
in  1^  tenths.)    in  .2^  ? 

3.  How  many  hundredths  in  .0^  ?  in  .01^  ?  in  .0^  ?  in 
.04? 

4.  How  many  thousandths  in  1  ?    in  .1  ?  in  .01  ?  in 
•OH? 

5.  How  many  thousandths  in  .01  ?   in  .06  ?    in  .05|  ? 

6.  How  many  tenths  in  1  ?   in  1^  ?    in  If  ? 

7.  How  many  hundredths  in  3  ?   in  3.1  ?   in  3.7  ? 

8.  How  many  thousandths  in  4  ?   in  4^  ?    in  4.5  ? 

9.  Review  and  write  answers  to  each. 

Each  of  the  following  equals  how  many  thousandths  ? 

1.  .1.  4.2.  2. 

2.  .7.  .5.  .04. 

3.  .3.  6.42.  3.7. 

4.  .04.  24.35.  .26. 

Each  of  the  following  equals  how  many  ten  thousandths  ? 

1.  .8.  4.5.  .04. 

2.  .85.  9.  4. 

3.  .164.  33.75.  .295. 


ADVANCED    ARITHMETIC. 


57 


1.  f  is  the  ratio  of  what  to  24  ?  What  is  the  simplest 
expression  of  the  ratio  of  18  to  24  ? 

2.  Jf  is  the  ratio  of  wliat  to  27  ?  What  is  the  simplest 
expression  of  the  ratio  of  15  to  27  ? 

3.  .15  is  the  ratio  of  what  to  100  ?  .15,  or  ^i/^  =  ^\. 
What  is  the  ratio  of  3  to  20  ?  What,  then,  is  the  ratio  of 
15  to  100  ? 

Express  the  following  ratios  in  the  simplest  form  — 

4.  .75.  3.5.  37.25.  27.4000. 

5.  .125.  8.94.  .875.  28.0004. 

6.  .225.  4.75.  .375.                     .075. 

7.  .35.  6.25.  .625.  28.3. 


1.    What  is  f  of  19  ? 


What  is  J  of  15  ? 


3.8 


3     X^ 


=  11.4. 


1  =  2.333^. 


The  4  equals  how  many  tenths  ? 
1  of  4.0  equals  what  ? 
What,  then,  is  i  of  19  ? 
What,  then,  equals  f  of  19  ? 

2.  What  is  -J-  of  7,  expressed  by  3  decimal  places  ? 
■J  of  6  equals  what  ? 
■J  of  .9  equals  what  ? 
■J  of  .09  equals  what  ? 

i  of  .010  equals  what  ? 
What,  then,  is  ^  of  7  ? 

3.  If  a  is  1,  what  is  b?     What  is  the  relation  of  -J  of  2 
to  ^  of  1  ?    Then  J  of  2  equals  how  many 
J  of  1  ?     Then  t  of  1  equals  i  of  what  ? 

What  is  the  relation  of  :|-  of  3  to  |-  of 
1  ?  Then  :i-  of  3  equals  how  many  ^  of 
1  ?     Then  f  of  1  equals  i  of  what  ? 

f  of  12  equals  i  of  what  ?  i  of  3  *  12's 
equals  f  of  what  ?  What  is  the  ratio  of  | 
of  12to  Jof  3-12's? 


58  ADVANCED    ARITHMETIC. 

f  of  7  equals  -^  of  how  many  7's  ?  f  of  100%  equals  ^ 
of  how  many  100%?  f  of  100  apples  equals  i  of  how 
many  100  apples  ? 

4.  Review  pp.  245,  246,  "Elementary  Arithmetic.'' 

5.  What  is  the  ratio  of  f  of  1  to  J  of  3  ? 

6.  What  is  the  ratio  of  }  of  5  to  |  of  1  ? 

7.  Which  weighs  the  more,  |  of  1  lb.  or  J  of  4  lb.  ? 

8.  What  is  f  of  1  or  ^  of  5  ? 

-J  of  4.8  equals  what  ? 
.625     ^  of    .16  equals  what  ? 
8 1 5.000'    J  of    .040  equals  what  ? 

What,  then,  is  J  of  5  or  |  of  1  ? 

9.  Express  5f  decimally. 

3  equals  how  many  tenths  ? 
i  of  2.8  equals  what  ? 
-ji  _  5  75     i  of  .20  equals  what  ? 

^  ~    '     '    What,  then,  is  i  of  3,  or  f  of  1,  expressed  deci- 
mally ? 
What,  then,  is  5f  expressed  decimally  ? 

Express  each  of  the  following  decimally.  Do  not  carry 
the  expression  to  more  than  3  decimal  places. 


10.    2^. 

14. 

ii. 

18. 

12i. 

22.    f. 

11.    X 

15. 

6|. 

19. 

7f. 

23.    if. 

12.    6i. 

16. 

% 

20. 

^V 

24.    16f 

13.   4f. 

17. 

H- 

21. 

37i. 

25.    62i 

State  the 

ratio  of 

0^  to  ^  and  of  ^  to  a. 

a 

b 

a 

b 

1.    100. 

25. 

100. 

16f. 

2.    100. 

75. 

100. 

.      33i. 

3.    100. 

20. 

100. 

66f. 

4.    100. 

40. 

100. 

83i. 

5.    100. 

60. 

100. 

12i. 

ADVANCED    ARITHMETIC.  59 


1.    What  is  the  sum  of  25  •  84  ? 
4)  8400     What  is  the  sum  of  100  '  84's  ? 


2100     What,  then, 

is  the 

sum  of  25 

-84's? 

Find  the  sum  of  — 

2.    25-648. 

8. 

16f  425.79. 

3.    75-9.85. 

9. 

83^-947. 

4.   331-8.45. 

10. 

12i  •  57.88. 

5.    66|-75. 

11. 

37i-  •  77.45. 

6.    60-78.45. 

12. 

87i-- 77.45. 

7.    80-64. 

13. 

250 • 7853. 

1.  At  $38^  an  acre,  what  is  the  cost  of  9.47  acres  of 
land  ? 

315.66f     At  $100  an  acre,  what  does  the  land  cost  ? 
3)947  What,  then,  is  the  cost  at  $33-J  an  acre  ? 

2.  What  is  the  cost  of  9.47  acres  of  land  at  $25  an 
acre?   at  $37^? 

3.  At  $1  a  box  a  grocer  paid  $200  for  fruit.     What 
would  it  have  cost  at  $.75  a  box  ?   at  $.60  ? 

4.  At  $1  a  bushel,  what  is  the  cost  of  160  bushels  of 
flax?   at  $.87^  a  bushel? 

5.  What  is  the  ratio  of  the  cost  of  government  land  at 
$1.83-J  an  acre  to  the  cost  at  $1  an  acre  ? 

6.  If  m  equals   .66f  of  the  cost  of  an  article,  what 
equals  the  cost  ? 

7.  What  is  the  ratio  of  a  rectangle  to  .8  of  it  ? 

8.  There  are  37  sq.  in.  in  .9  of  a  rectangle.    How  many- 
square  inches  in  the  rectangle  ? 

9.  If  .7  of  the  distance  between  two  places  equals  168 
miles,  what  is  the  distance  ?     Show  by  drawing. 

10.    2400  miles  equals  .7  of  the  distance  between  two 
places.     What  is  the  distance  ?     Show  by  drawing. 


60  ADVANCED    ARITHMETIC. 

11.  A  merchant  buys  caps  for  $1  and  sells  for  $1.62^. 
The  gain  equals  what  part  of  the  cost  ? 

12.  .12|-  of  the  money  a  man  invested  equaled  his 
profit.  If  $600  was  the  profit,  what  equaled  the  amount 
invested  ? 

13.  A  man  bought  a  horse  for  $x  and  sold  it  for  $180. 
The  selling  price  equaled  |  of  $x.  What  was  the  cost  of 
the  horse  ? 

14.  What  part  of  the  cost  of  hats,  at  $1  each,  must  be 
added  to  the  cost  to  find  the  cost  at  $1.33^  each  ?  at 
$1.37^  each?   at  $1.87^  each?   at  $1.62^  each? 

15.  $160  is  the  cost  of  apples  at  $1  a  bushel.  If  the 
same  number  of  bushels  cost  the  sum  of  $160  and  i  of 
$160,  what  is  the  price  per  bushel? 

16.  Carpeting  which  cost  $1  a  yard  is  sold  for  87^/  a 
yard.     The  loss  equals  what  part  of  the  cost  ? 

17.  A  furniture  dealer  paid  $100  for  chairs  at  $1  each. 
He  sold  them  for  J  more  than  they  cost  him.  What  was 
the  selling  price  of  each  chair  ?  What  did  he  receive  for 
all? 

18.  A  milliner  bought  hats  at  $1  and  marked  them  to 
sell  for  ^  more  tlian  they  cost.  What  was  the  marked 
price  ? 

She  sold  them  for  ^  less  than  the  marked  price.  What 
did  she  receive  for  them  ?  Show  by  drawing.  Did  she 
make  or  lose  by  buying  and  selling  the  hats  ? 

19.  A  merchant  bought  dress  goods  at  75/  a  yard,  and 
marked  it  to  sell  at  20%  profit,  but  sold  it  at  10%  below 
marked  price.     Did  he  make  or  lose  by  the  transaction  ? 

20.  At  40/  a  pound,  what  is  the  cost  of  5  lb.  12  oz.  of 
butter  ? 

21.  What  is  the  cost  of  4  pk.  and  7  qt.  of  cherries  at  $1 
a  peck  ? 

22.  At  $2  a  bushel,  what  is  the  cost  of  3  pk.  of  plums  ? 


ADVANCED    ARITHMETIC.  61 

1.  If  10  weighs  18,  what  does  17  weigh  ? 
What  are  1.7  (|^)  of  18  ? 

What  is  .1  of  18  ?  Ans.    1.8    (eighteen  tenths). 

What,  then,  are  1.7  of  18  ?     17  •  1.8  =  30.6. 

2.  What  is  .25  of  .75  ? 

.75  _        What  is  a  simpler  expression  than  ^^^^  ? 
T  ~  •      What  is  i  of  .75  ? 

3.  Review  pp.  243,  244,  "  Elementary  Arithmetic." 

4.  What  is  2.48  (f  §-«)  of  2.75  ? 


62       .11                 What 

is  2V 

of  2.75  ? 

U^  •  t.li^                  What 
100                       What, 

5.      .25  of  .36  ? 

is  i  of  248  ? 
then,  is  ^io  of  248  '  2.75  ? 

12.        .3  of  750  ? 

6.      .5  of  68  ? 

13.      1.2  of  3.12  ? 

7.    3.2  of  70  ? 

14.    24.6  of  98  ft.? 

8.      .75  of  2.05  ? 

15.      2.3  of  6.25  mo.  ? 

9.      .39  of  .875  ? 

16.        .27  of  $98.75  ? 

10.      .04  of  6.5  ? 

17.    453.  of  $5.75? 

11.      .025  of  823? 

18.    Y- of  ^4.56? 

1.  What  is  the  cost  of  52.75  bu.   of  oats  at  35)2^  a 
bushel  ? 

2.11  At  $1  a  bushel,  what  is  the  cost  of 

35  •  n.1t$  ^  ^g  ^g  1  the  oats  ? 

100  '     *  '     What,  then,  is  the  cost  at  35/  ? 

4 

2.  What  is  the  cost  of  16.5  yd.  of  muslin  at  8/  a  yard  ? 
3. .  What  does  a  man  earn  in  15.5  days  at  $2.65  per  day  ? 

4.  Wliat  is  the  cost  of  12,200  shingles  at  $4.80  per 
1000? 

5.  What  is  the  cost  of  10,240  bricks  at  $5.25  per  1000  ? 

6.  What  is  the  cost  of  650.75  bu.  of  flax  at  $1.46  per 
bushel  ? 


62  ADVANCED    ARITHMETIC. 

7.  What  is  the  cost  of  2600  envelopes  at  $2.16  per  100  ? 

8.  At  65^  per  pound,  what  must  be  paid  for  254.75  lb. 
of  tea  ? 

9.  At  $5.35  per  ton,  what  must  be  paid  for  12.5  tons 
of  coal  ? 

10.  What  is  the  cost  of  14.5  yd.  of  calico  at  6:J^  a  yard  ? 

11.  A  man  sold  79.25  cords  of  wood  at  $3.08  a  cord. 
How  much  did  he  receive  ? 

12.  What  is  the  cost  of  675  books  at  $1.12^  each  ? 

1.    What  is  .Of  of  824  lb.  ?     {Read:  What  is  |  of  .1  of 

824  lb.  ?) 

20.6  What  is  .1  of  824  ? 

3  •  n-^  _^.  ^      What,  then,  is  .Of  ? 
—  bl.o. 


2.  What  is  .OOf  of  824  lb.  ? 

3.  What  is  .Of  of  $630  ? 

4.  What  is  .Of  of  84-i%  ? 

5.  What  is  .Of  of  .24  ? 

6.  What  is  .2^  of  96  ? 

2.4  What  is  .1  of  96  ? 

^  '  ^-^  _  oi  ^      What,  then,  is  .Of  of  96  ? 

7.  What  is  .2^  of  56.7  ? 

8.  What  is  .2f  of  8.49  ? 

9.  If  a  barrel  of  apj^les  costs  $5.75,  what  is  the  cost  of 
5.3  bbl.  ? 

10.  What  is  the  value  of  .2^  acres  of  land  at  $37.50  ? 

11.  If  .05  of  a  farm  is  worth  $472,  what  is  the  farm 
worth  ? 

12.  If  824  lb.  is  .Of  of  the  wheat  in  a  bin,  what  is  the 
amount  of  wheat  in  the  bin  ? 

13.  %  oi  X  equals  J  of  what  ? 
What  part  of  x  equals  J  of  2  a;  ? 


ADVANCED    ARITHMETIC. 


63 


1.  If  .02  is  the  ratio  of  a  to  h,  what  is  the  ratio  of 
^  to  a? 

What  is  the  ratio  of  a  to  the  sum  of  a  and  Z>  ?  of  ^  to 
the  sum  ?  of  the  sum  to  a  ?  of  the  sum  to  6  ?  of  a  to  their 
difference  ?   of  5  to  their  difference  ? 

2.  What  is  the  ratio  of  .07  to  3.5  ? 

1  3.5  equals  how  many  hundredths  ? 

1        ^^     What  is  the  ratio  of  7  hundredths  to  350 


300      50 
50 


=  .02. 


hundredths  ? 
What,  then,  is  the  ratio  of  .07  to  3.5  ? 


3.  If  .02  is  the  ratio  of  .07  to  3.5,  what  is  the  ratio  of 
3.5  to  .07  ? 

4.  What  is  the  ratio  of  3.5  to  .07  ? 

What  is  the  ratio  of  350  hundredths  to  7 

hundredths  ? 
What,  then,  is  the  ratio  of  3.5  to  .07  ? 
Why  does  the  ratio  of  350  hundredths  to  7 
hundredths  equal  the  ratio  of  3.5  to  .07  ? 
What  is^the  ratio  of  12.5  to  .45  ? 

What  is  the  ratio  of  1250  hundredths 

—  =  27  777         *^  ^^  hundredths  ? 
0  9  '       '    What,  then,  is  the  ratio  of  12.5  to 

9  .45? 

Why  does  the  ratio  of  1250  to  45  equal  the  ratio  of  250 
to  9? 


50 

7^350  =  50. 


5 
250 


64  ADVANCED    ARITHMETIC. 

6.    What  is  the  ratio  of  .45  to  12.5  ? 
.036 


1250)45.00  What  is  the  ratio  of  45  hundredths  to  1250 
3750  hundredths  ? 

7500  What,  then,  is  the  ratio  of  .45  to  12.5  ? 
7500 


What  is 

the  ratio  of  a 

to  h  and  of  ^  to  «  ? 

a 

h 

a 

h 

.95. 

1.9. 

6. 

42. 

.08. 

.4. 

52.5. 

.015. 

8.12. 

4. 

.0256. 

.27. 

.25. 

7.5. 

3. 

.4. 

1.5. 

1.05. 

24. 

.0001. 

682.5. 

25. 

$16. 

$.06i. 

.015. 

.5. 

$12.62f 

$5. 

682.5. 

25. 

$16. 

$.06i. 

45.825. 

150. 

$5000. 

$.125. 

.01. 

4. 

$75. 

$1.25. 

15.77. 

.083. 

$67.83. 

$.75. 

3.56. 

3.9. 

$.875. 

$.12^. 

6.25. 

.05. 

135.05. 

.327. 

625. 

.05. 

17.28. 

1728. 

786. 

5. 

135. 

.37i. 

.05. 

64.5. 

$22.10. 

1.70. 

625. 

.08. 

1.70. 

22.10. 

In  each  of  the  above,  express  the  ratio  of  a  to  h  and  h  to 
a  in  hundredths,  thus  : 

95  j.^      .50  is  the  ratio  of  .95  to  1.9  expressed  in 

—  =  ^  =  .oO.         i,undredths. 

190  _  2.00  is  the  ratio  of  1.9  to  .95  expressed 

95  *     '  in  hundredths. 


ADVANCED    ARITHMETIC.  65 

1.  If  two  shovels  are  worth  $1.70,  how  many  can  be 
bought  for  $22.10? 

2.  If  26  shovels  cost  $22.10,  how  many  can  be  bought 
for  $1.70? 

3.  If  a  dozen  roses  can  be  bought  for  87|^/,  how  many 
dozen  can  be  bought  for  $10.50  ? 

4.  At  $2.50  per  100,  how  many  100  bananas  can  be 
bought  for  $40? 

5.  If  you  buy  bananas  at  $2.50  per  100,  and  sell  them 
for  20/  more  than  cost,  what  do  you  receive  for  the 
bananas  ? 

6.  If  oranges  are  35/  a  dozen,  how  many  can  be  bought 
for  $10.70  ? 

7.  At  $1.75  per  yard,  how  many  yards  of  silk  can  be 
bought  for  $18.20  ? 

8.  If  a  half  dozen  dress  patterns,  each  containing  7  yd., 
sell  for  $68.25,  what  is  that  per  yard  ? 

What  is  the  ratio  of  2^  to  3^  expressed  in  hun- 
dredtlis  ? 

i|  is  the  ratio  of  2^  to  3^  expressed  as  a  common  frac- 
tion,    jo  =  .76i§. 

Express  the  relation  of  a  to  h  in  hundredths  : 


a     - 

h 

a 

b 

7. 

2h 

2i. 

.3*. 

i- 

h 

h 

*• 

37i.       ^ 

5. 

H- 

•H- 

5i. 

6i. 

h 

A- 

5. 

5i. 

i- 

h 

.75. 

.6. 

6.22. 

5.6. 

Express  in  hundredths  : 

r§'  ^%'  TU'  f'  h  t'  8")  i)  |j  T'  i?  tV'   t\>   tVj    t^&>    A- 


66  ADVANCED    ARITHMETIC. 

1.  7  is  the  ratio  of  what  unit  to  $9  ?  What  is  the  ratio 
of  $63  to  $9  ?     $9  equals  what  part  of  $63  ? 

2.  f  is  the  ratio  of  what  unit  to  45  ?  What  is  the  ratio 
of  45  to  27  ?  27  equals  what  part  of  45  ?  If  is  the  ratio 
of  what  unit  to  27  ? 

3.  .07  is  the  ratio  of  what  unit  to  $600  ?  What  is  the 
ratio  of  7  •  $6  to  $600  ?  7  *  $6  equals  how  many  hundredths 
of  $600  ?     $600  equals  how  many  7ths  of  7 '  $6  ? 

What  is  the  ratio  of  the  wheat  that  can  be  bought  for 
$600  to  the  wheat  that  can  be  bought  for  7  *  $6  ? 

The  wheat  that  can  be  bought  for  7  *  $6  equals  what  part 
of  the  wheat  that  can  be  bought  for  $600  ? 

4.  What  is  .00^  of  $8400  ?  (Read:  What  is  ^  of  ^-J^ 
of  $8400  ?)  What  is  .01  of  $8400  ?  What,  then,  is  .00^ 
of  $8400? 

5.  What  equals  .OOf  of  754  lb.  ?  (What  is  f  of  ji^  of 
754  lb.?  What  is  .01  of  754  lb.?  What,  then,  is  .OOf 
of  754  lb.? 

6.  What  equals  .02|  of  $24.56  ? 
1.885 


3 


4 


5.655. 


1.  What  is  the  ratio  of  .03^  to  .01  ? 

Ans.    |-  is  the  ratio  of  .03|-  to  .01. 

2.  State  the  ratio  of  each  of  the  following  to  .01 :  .03|, 
.021,  .031,  .09i,  .17i,  .101,  .06|. 

3.  What  is  the  ratio  of  .01  to  .03^-  ? 

Ans.    I  is  the  ratio  of  .01  to  .03^. 

4.  State  the  ratio  of  .01  to  each  of  the  following  :  .03|, 
.021,  .03f,  Mh  .17i,  .07f,  .061,  .04f,  .OOf,  .00^,  .OOf, 
.004i. 

,  5.    What  is  the  ratio  of  .01  to  each  of  the  following  ? 
.7,  .21,  .9f,  .007,  .33^,  3.7. 


ADVANCED   ARITHMETIC.  67 


1.  What  is  .OOf  of  $824  ? 

3-8.24  __  ^  What  is  .01  of  $824  ? 

4      ""  *  What,  then,  is  .OOf  of  $824  ? 

2.  What  is  .OOf  of  $2114  ? 

3.  What  is  .00|  of  $840  ? 

4.  What  is  .OOf  of  947  ft.  ? 

5.  What  is  .OOf  of  875  da.  ? 

6.  What  is  .02^  of  78  tons  ? 

5  •  .78  _  ^  What  is  .01  of  78  tons  ? 


2  •               What,  then,  are  .02^  of  78  tons  ? 

7.  What  is  .02^  of  $729  ? 

8.  What  is  .07^  of  16.72  bn.  ? 

9.  What  is  .13|  of  5764  oz.  ? 

1.  What  is  the  ratio  of  a  unit  to  .07  of  the  unit  ? 

$255  is  .07  of  what  unit  ? 

^  =  ?     What  equals  .01  of  the  unit  ? 

^  What,  then,  equals  |g§  of  it  ? 

2.  96/  equals  .3  of  what  ? 

3.  24  ft.  equals  .25  of  what  ? 

4.  645  pk.  equals  .60  of  what  ? 

5.  231  gal.  equals  .03^  of  what  ? 

What  is  the  ratio  of  .01  to  .03^  ? 
100  2   231  __  ^     .01  of  the  unit  equals  what  part  of  231 
7  "•         gal.? 

The  unit  equals  what  ? 

6.  $846  are  .01|  of  what  unit  ? 

7.  854.37  aje  .5^  of  what  unit  ? 
8..  247  yd.  are  .02f  of  what  ? 

9.  $675.25  are  .02f  of  what  ? 

10.  $723.54  are  .OOf  of  what  ? 

11.  68  ft.  are  .OOf  of  what  ? 

12.  2745  are  .00|  of  what  ? 

13.  32.3  are  .Of  of  what  ? 


68  ADVANCED    ARITHMETIC. 

1.  When  a  ball  is  thrown  into  the  air,  what  force  pulls  it  to 
the  earth  ? 

2.  Name  things  upon  which  this  force  acts. 

3.  What  can  you  name  that  is  not  influenced  by  the  force  of 
gravity  ? 

4.  When  a  boy  slips  and  falls,  what  pulls  him  to  the  floor? 

5.  What  holds  the  oceans,  lakes,  air,  etc.,  in  their  places  ? 

6.  When  a  dredging  boat  is  unloaded,  what  pulls  the  mud, 
sand,  and  stones  through  the  water? 

7.  What  pulls  rain  and  snow  through  the  air  to  the  surface  of 
the  earth  ? 

8.  If  you  place  a  stick  at  the  bottom  of  a  vessel  of  water,  what 
force  pushes  it  to  the  surface  of  the  water  ? 

9.  If  a  boy  uses  a  pole  to  raise  a  window,  is  the  boy  or  the 
stick  the  primary  cause  of  the  movement  of  the  window  ? 

10.  What  force  pushes  an  ascending  balloon  away  from  the 
surface  of  the  earth  ? 

11.  If  a  6-in.  cube  of  wood  weighs  as  much  as  a  6-in.  cube  of 
water,  will  the  6-in.  cube  of  wood,  if  placed  in  a  tub  of  water,  sink 
to  the  bottom?  Why?  If  the  cube  of  wood  is  placed  at  the 
bottom  of  the  tub,  will  it  rise  ? 

12.  Why  does  a  stone  fall  through  water?  Why  does  a  snow- 
flake  fall  through  the  air  ? 

13.  What  is  true  of  the  pulling  power  of  gravity  upon  a  stone 
and  upon  an  equal  volume  of  water  ?  upon  a  snowflake  and  an 
equal  volume  of  air  ? 

14.  What  causes  some  soap-bubbles  to  rise?  What  is  in  a 
soap-bubble  ? 

What  is  the  difference  between  the  air  in  a  soap-bubble  and  the 
air  about  it  ? 

15.  What  is  weight  ? 

16.  Which  is  heavier,  a  floating  balloon  or  a  descending  snow- 
flake? 

17.  Throw  a  ball  into  the  air.  While  rising,  does  it  Iiave 
weight  ? 

18.  In  which  is  an  apple  the  heaviest,  in  water,  in  air,  or  in  a 
vacuum  ? 


ADVANCED    ARITHMETIC.  69 

Specific  Gravity.  —  The  specific  gravity  of  any  substance 
is  the  ratio  of  the  weight  of  the  substance  to  the  weight  of 
an  equal  volume  of  water. 

The  following  is  the  specific  gravity  of  a  few  liquids  and 


solids  : 

Ice 

.93. 

Alcohol                      .791. 

Iron  (wrought) 

7.6  to  7.8. 

Quicksilver            13.596. 

Silver 

10.5. 

Milk                        1.032. 

Gold 

19.  to  19.6. 

Anthracite    coal     1.8. 

Tin 

7.29. 

Bituminous     "       1.25. 

Oak 

.84. 

Lead                      11.35, 

1.  How  many  ounces  in  a  cubic  foot  of  water'  if  it 
weighs  62^  lb.? 

2.  What  part  of  1000  oz.  does  a  cubic  foot  of  ice 
weigh  ?  Why  ?  How  many  pounds  does  a  cubic  foot 
of  ice  weigh  ? 

3.  What  is  the  weight  of  a  6-in.  cube  of  silver  ? 

4.  What  is  the  weight  of  a  gold  brick  2  in.  by  4  in.  by 
8  in.  ? 

5.  If  a  cubic  foot  of  water  weighs  1000  oz.  and  a  cubic 
foot  of  quartz  2650  oz.,  what  is  the  specific  gravity  of  the 
quartz  ? 

6.  How  many  cubic  inches  in  75  oz.  of  alcohol  ? 

7.  What  is  the  specific  gravity  of  a  bar  of  iron  4  in. 
square  and  10  in.  long,  if  it  weighs  40  lb.  ? 

8.  If  a  substance  heavier  than  water  is  immersed  in 
water,  the  water  buoys  it  up  just  the  amount  of  the  weight 
of  the  water  the  substance  displaces.  If  a  cubic  foot  of 
stone  weighs  1500  oz.  in  water,  what  is  the  specific  gravity 
of  the  stone  ? 

9.  If  just  ^  of  a  log  of  wood  floats  above  water,  what 
is  its  specific  gravity  ? 


70 


ADVANCED    ARITHMETIC. 


1.  What  ratios  do  you  see  ? 

2.  If  h  is  100,  what  is  a?     If  ^>  is  100%   (per  cent), 
what  is  a? 

3.  Make  similar  sentences  :     If  100%  is  6,  50%  is  3. 

4.  3  equals  what  %  of  6%  ?     -J-  equals  what  %  of  |  ? 

5.  If  a  is  50%,  what  is  the  sum  of  a  and  h  ? 

6.  If  150%  is  12,  what  is  100%  ?  50%  ? 

7.  If  150%  is  15,  what  is  100%  ? 

8.  10  equals  what  %  of  10  ?     6  equals  what  %  of  6  ? 

9.  a  equals  what  %  of  6  ? 

10.  i  oi  b  equals  what  %  of  a  ?   of  J  ? 

11.  What  is  the  ratio  of  100%  to  50%?  of  50%  to 
100%  ?  What  part  of  100%  equals  50%  ?  If  ^>  is  100%, 
the  sum  of  a  and  b  equals  what  %  ? 

12.  The  difference  between  a  and  b  equals  what  %  of  6  ? 
of  a? 

13.  If  b  is  25%,  what  is  a  ?  If  ^»  is  10%,  what  is  a? 
If  a  is  |-%,  what  is  Z>  ? 

14.  This  is  50%.     Draw  a  rectangle  equal  to  100%. 

15.  This  line  is  100%.  Draw  a  line  equal  to  50%; 
to  200%  ;  to  150%. 


ADVANCED   ARITHMETIC.  71 

16.  Observe  things  in  the  room  and  write  five  state- 
ments similar  to  the  following:^  2  v^indows  equals  50% 
of  4  windows.  3  rows  of  desks  equals  50%  of  6  rows.  ^ 
the  blackboard  equals  50%  of  the  entire  blackboard. 

17.  Show  by  drawings  the  relative  magnitude  of  things 
whose  ratio  equals  that  of  50%  to  100%  ;  of  50%  to  the 
sum  of  50%  and  100%.    . 

18.  Write  five  statements  similar  to  the  following :  50% 
of  8  books  equals  4  books.  50%  of  the  coal  in  the  bin 
equals  ^  of  it.     50%  of  10  hooks  Equals  5  hooks. 

19.  50%  is  the  ratio  of  what  to  4  ?  to  6  ?  to  ^  ?  to  20  ? 
toi? 

20.  Write  five  questions  similar  to  the  following  :  A  boy 
had  12  marbles  and  lost  50%  of  them.  How  many  had  he 
left  ? 

21.  What  is  50%  of  each  of  the  following  units  ?  4  pk., 
8  ft.,  $20,  16  cu.  in.,  180  mi.,  1800  yr.,  $^,  75/. 

22.  Give  quickly  the  units  of  which  the  following  is 
50%  :  5  lb.,  12  bu.,  $120,  f,  60  ft.,  180  da.,  25/,  5  dimes, 
3i,  13  halves,  J,  $150,  75/. 

23.  5  lb.  equals  what  %  of  10  lb.  ?  i  equals  what  %  of 
2  ?    7  equals  what  %  of  3^  ? 

24.  The  ratio  of  50%  to  100%  equals  the  ratio  of  75/ 
to  what  ? 

25.  Express  the  ratios  of  a  to  b  and  ^  to  <x  thus :  6  qt. 
equals  50%  of  12  qt. 


a 

b 

a 

b 

8  qt. 

2  gal. 

f- 

f- 

25/. 

$h 

2|- 

5*. 

15  ft. 

10  yd. 

18  sq.  ft. 

12  sq.  ft. 

2  cu.  yd. 

54  cu. 

ft. 

1. 

«• 

1  The  form  equals  is  used  because  one  whole  is  to  be  thought  of  in 
relation  to  another  whole. 


72 


ADVANCED    AKITHMETIC. 


1.  What  ratios  do  you  see  ? 

2.  If  a  is  100,  what  is  6  ?  what  is  c  ? 

3.  If  a  is  100%,  what  ish?  what  is  c  ? 

4.  How  many  33^%  in  each  unit  ? 

5.  What  is  the  ratio  of  100%  to  33^%  ?  3  is  the  ratio 
of  100%  to  what  part  of  66|%  ? 

6.  What  is  the  ratio  of  66f  %  to  33  J  %  ?  to  f  of  100%  ? 

7.  Find  solids,  surfaces,  and  lines  respectively  having 
the  ratios  100%,  66§%,  and  33^%,  and  compare. 

8.  Observe  things  in  the  room  and  write  five  sentences 
similar  to  the  following :  1  box  of  chalk  equals  33^%  of 
3  boxes.  2  desks  equals  33^%  of  6  desks.  -J  of  the  floor 
equals  33^%  of  it. 

9.  Recall  units  and  write  statements  like  this  :  5  bu. 
equals  33J%  of  15  bu.  $25  equals  33^%  of  $75.  J  of 
an  acre  equals  33^%  of  an  acre. 

If  there  has  been  activity  of  mind  through  the  senses  in  the 
previous  work  ;  if  varied  experiences  have  brought  simple,  basic 
relations  before  the  pupil  under  different  ^  forms,  progress  will  be 

1  ' '  Only  after  there  have  been  received  many  experiences  which 
differ  in  their  kinds  but  present  some  relation  in  common,  can  the 
first  step  be  taken  towards  the  perception  of  a  truth  higher  in  gen- 
erality than  these  different  experiences  themselves."  —  Herbert 
Spencer. 


ADVANCED    ARITHMETIC.  73 

easy  and  rapid  at  this  stage.  Let  the  language  employed  be  given 
meaning  by  being  used  in  direct  ^  connection  with  things.  To  be 
obliged  to  translate  what  should  be  familiar  expressions,  instead 
of  at  once  realizing  them  in  thought,  is  a  waste  of  energy. 

10.  33  J  %  is  the  ratio  of  each  of  these  units  to  what  ? 
12  pk.,  $6,  400  bu.,  90  oz.,  35/,  18  houses,  9  miles. 

11.  Think  quickly  J  of  each  of  the  following :  $24,  15/, 
72  lb.,  $150,  300  sheep,  1800  books,  42  sq.  ft.,  1  cu.  yd. 
Think  33  J  %  of  each  of  the  above  units. 

1.2.  Observe  things  in  the  room  and  write  five  sentences 
like  these  :  66|%  is  the  ratio  of  2  hats  to  3  hats.  66§%  is 
the  ratio  of  6  slates  to  9  slates.  66|%  is  the  ratio  of  f  of 
the  west  wall  to  the  entire  wall. 

13.  What  is  66^%  of  live  quantities  that  you  observe  in 
the  room  ?     Ex.  66^^o  of  6  windows  is  4  windows. 

14.  What  is  660o  of  $18  ?  of  600  pt.  ?  of  1200  bu.  ? 

15.  $12  equals  what  %  of  $18  ?  400  pt.  equals  what  % 
of  600  pt.  ?     66 1  %  is  the  ratio  of  what  to  18  ? 

16.  What  is  the  ratio  of  $12  to  $18  expressed  in  %  ? 
Name  quantities  in  the  room  having  the  ratio  of  66^%. 
Ex.  ^^%^o  is  the  ratio  of  4  windows  to  6  windows. 


i 


17.  If  a  is  3%,  what  is  S  ?  c  ?  If  a  is  1%,  what  is  ^»  ? 
c  ?  If  a  is  i%,  what  is  ^  ?  c  ?  \ih\%  }^%,  what  is  c  ?  a? 
If  b  is  ic%,  what  is  c  ?  a? 

18..  2  pt.  equals  what  %  of  3  qt.  ?  3  pk.  of  1^  bu.  ? 
$1.50  of  $1.00  ?  3  sq.  yd.  of  3  yards  square  ?  33  ft.  of  11 
yd.  ?  5  ft.  of  5  yd.  ?  x  ft.  of  x  yd.  ? 

1  u  Words  cannot  attain  definiteness  save  as  living  outgrowths  of 
realities,  as  the  exact  expression  of  the  phenomena  of  life." — Dr. 
Maudsley. 


74  ADVANCED   ARITHMETIC. 

Place  prisms  having  the  ratios  of  the  units  1,  2,  3,  and  4 
where  they  can  be  obsei-ved. 

1.  What  ratios  do  you  see  ? 

2.  If  a  is  100,  what  is  each  of  the  other  units  ?     If  a 
is  100%,  what  is  each  of  the  other  unit's  ? 

3.  Tell  the  ratios  of  these  different  %'s. 

4.  What  is  the  ratio  of  2^%  to  50%?  to  75%?  to 
100%? 

5.  What  is  the  ratio  of  each  unit  to  each  of  the  others 
expressed  in  %  ? 

6.  Make  similar  sentences  :  33-J-%  is  the  ratio  of  d  to 
the  sum  of  d  and  c. 

7.  Introduce  exercises  similar  to  those  in  the  preced- 
ing lessons  to  familiarize  the  ratios  of  the  above  %'s. 

8.  Make  problems. 

Make  similar  sentences:  1  is  the  ratio  of  75%  to  |  of 
100%;  to  I  of  50%  ;  to  3  times  25%.  2  is  the  ratio  of 
50%  to  1  of  100%. 

9.  Call  c  10  and  name  the  others.      Give  the  per  cent 
relation  of  10  to  5 ;  to  15 ;  to  20. 

10.  If  a  is  4%,  what  is  each  of  the  others  ?  If  a  is  1%, 
what  is  each  of  the  others?  If  d  is  ^%,  what  is  each  of 
the  others  ? 

11.  Of  what  are  the  following  25%?  |,  35  pencils, 
$1.25,  2f ,  -y- 

12.  35/  equals  what  %  of  $1.40? 

What  is  the  ratio  of  13  ft.  to  26  ft.  ?  of  27  da.  to  108  da.  ? 
of  3  wk.  to  28  da.  ?  of  |  to  If  ?  of  Ij  to  f  ?  of  J  in.  square 
to  1  in.  square  ?  of  f  to  2  ic  ?  In  each  what  %  is  the 
latter  of  the  former  ? 

Review  pp.  286-294  inclusive,  "  Elementary  Arithmetic." 
Draw  rectangles  on  the  blackboard  having  the  ratios  of  the 
units  1,  2,  3,  4,  5. 


ADVANCED   AKITHMETIC. 


75 


a  b  c  d  e 

1.  What  ratios  do  you  see? 

2.  If  d  is  100%,  what  is  each  of  the  other  units? 

3.  Compare  each  with  each  of  the  other  four. 

4.  If  6  is  100%,  what  is  each  of  the  other  units? 

5.  Compare  each  with  each  of  the  others.  Compare 
without  observing  the  units.  If  e  is  1%,  what  is  each  of 
the  others?     Ifeis5%? 

6.  Draw  the  rectangles  of  which  this  is 
50%;  25%;  75%;  20%;  60%;  40%;  80%; 
33i%;  66|%. 

7.  The  ratio  of  25%  to  75%  equals  the 
ratio  of  33-J%  to  what?     Show  by  drawing. 

8.  The  ratio  of  a  farm  to  66f  %  of  the  farm  equals 
the  ratio  of  what  part  of  the  farm  to  40%  of  it?  Show 
by  drawing. 

9.  If  100%  is  ^30,  what  is  40%  ?  80%  ?  20%  ? 

10.  If  75/  is  100%,  what  is  60%?  80%?  40%  ?  20%? 

11.  If  185  bu.  is  100%,  what  is  60%  ?  80%  ?  20%  ? 

12.  If   80%   is  one  ton,  20%   equals  how  many  lb.? 
80%? 

13.  If  60%  is  1  sq.  yd.,  100%  is  how  many  sq.  ft.  ? 

14.  How  much  is  40%  of  40  acres  ?  100%  of  40  acres? 
40%  of  $250  ?  of  1  sq.  in.  ?  of  2^  sec.  ?  of  ^  lb.  ?  of  J  pt.  ? 

15.  X  equals  the  cost  of  a  book  which  sells  for  20% 
above  cost.     Draw  figures  having  the  ratio  of  the  selling 


76  ADVANCED    ARITHMETIC. 

price  to  the  cost.  If  the  cost  of  a  book  is  $J,  what  is  the 
selling  price  ?  |  equals  how  many  5ths  of  J  ?  §  equals 
what  %  of  J?  The  difference  between  f  and  J  equals 
what  %  of  i  ? 

16.  At  20%  gain  my  profit  is  5/.  What  is  the  cost,  or 
100%  ?     What  is  the  selling  price? 

17.  If  75%  of  an  acre  equals  120  sq.  rd.,  50%  of  an 
acre  equals  how  many  sq.  rd.  ? 

18.  X  equals  50%  of  what  ?  66|%  of  what  ? 

19.  i-  of  a  blackboard  equals  what  %  of  |^  of  it  ? 

20.  A  2-in.  square  equals  what  %  of  a  4-in.  square  ? 

21.  The  perimeter  of  a  2-in.  square  equals  what  %  of  the 
perimeter  of  a  4-in.  square  ? 

22.  The  perimeter  of  a  rectangle,  1X2,  equals  what 
%  of  the  perimeter  of  a  2-in.  square  ? 

23.  The  perimeter  of  a  1-in.  square  equals  what  %  of 
the  perimeter  of  a  2-in.  square  ? 

24.  60%  of  40  in.  equals  how  many  ft.  ? 

25.  A  farmer  had  150  sheep.  He  sold  40%  of  them. 
How  many  had  he  left  ? 

26.  A  coat  cost  $15,  and  a  vest  66f  %  as  much.  What 
was  the  cost  of  both  ? 

27.  The  area  of  a  1-in.  square  equals  what  %  of  the 
area  of  a  2-in.  square  ? 

28.  40%  of  a  section  of  land  equals  256  acres.  How 
many  acres  in  the  section  ? 

29.  1200  lb.  equals  60%  of  a  ton.  How  many  lb.  in  the 
ton? 

30.  If  X  equals  80%,  what  equals  100%  ? 

31.  What  is  the  ratio  of  the  gain  to  the  cost  if  a  dealer 
buys  hats  for  $4  and  sells  them  for  $5  ?  if  he  buys  for 
$5  and  sells  for  $6  ?  if  he  buys  for  $3  and  sells  for  $4  ? 
if  he  buys  for  50/  and  sells  for  $1.00  ? 

32.  A  boy  bought  a  knife  for  80/  and  sold  it  at  a  profit 


ADVANCED    ARITHMETIC. 


77 


of  20%;  for  how  much  did  he  sell  it  ?  What  is  the  ratio 
of  the  cost  to  the  selling  price  ?  of  the  selling  price  to 
the  cost  ? 

33.  If  the  ratio  of  $40  to  the  cost  of  a  horse  is  20%, 
what  is  the  cost  of  the  horse  ? 

34.  If  in  a  district  of  2500  persons  the  number  of  chil- 
dren attending  school  is  500,  what  %  of  the  population 
attends  school  ? 

35.  If  B's  property  is  100%  and  A's  is  40%  more,  what 
%  is  A's  ?  What  is  the  ratio  of  B's  property  to  A's  ? 
Of  A's  to  B's  ? 

36.  John  sold  a  sled  for  $16,  which  was  60%  more  than 
the  cost.  What  was  the  ratio  of  the  selling  price  to  the 
cost  ?  of  the  cost  to  the  selling  price  ?  What  was  the  cost  ? 
If  he  had  sold  it  for  60  %  less  than  the  cost,  for  how  much 
would  he  have  sold  it  ? 

Place  prisms  having  the  ratios  1,  2,  3,  4,  5,  6  where  they  can 
be  observed. 


1.  If  a  equals  100%,  what  does   each   of  the   others 
equal  ? 

2.  Make  sentences  similar  to  these:  The  sum  of  16§% 
and  33i%  equals  50%.     83^%  is  16|%  less  tlian  100%. 

3.  Each  unit  equals   wliat    %    of   each   of   the   other 
units  ?     e  ec^uals  200%  of/,  66 1  of  d,  etc. 


T8  ADVANCED    ARITHMETIC. 

4.  Introduce  a  variety  of  objects  in  which  the  above 
relations  may  be  discovered. 

5.  22  is  what  %,  if  132  is  100%  ?  22  equals  what 
%  of  132  ?    of  88  ? 

6.  40  is  what  %,  if  48  is  100%  ?  40  equals  what  %  of 
48? 

7.  Find  66f  %  of  5  yd. ;  8  ft.  ;  1  cu.  yd.  Find  16f  %  ; 
83J%. 

8.  Make  and  answer  similar  problems  :  When  83  J  %  is 
15,  what  is  100%  ?  15  equals  83^%  of  what  ?  12  equals 
how  many  eighths  of  48  ?  80  equals  how  many  hun- 
dredths of  800  ?     4  equals  what  %  of  24  ? 

9.  What  is  the  ratio  of  33  J  %  to  50%  ? 
-When  50%  is  96,  what  is  331%  ? 

10.  50%  of  the  weight  of  a  bbl.  of  flour  is  98  lb.  33^% 
of  the  barrel  equals  what? 

11.  The  cost  of  a  book  is  40/  and  the  cost  of  a  tablet 
8/.  The  cost  of  the  tablet  equals  what  %  of  the  cost 
of  the  book  ? 

12.  John  had  72/.  He  gave  83^%  of  his  money  to  a 
blind  man.     How  much  had  he  left? 

What  is  the  ratio  of  the  amount  left  to  the  amount 
given  ? 

13.  A  2-ft.  square  equals  what  %  of  2  sq.  ft.  ? 

14.  The  surface  of  a  2-in.  cube  equals  what  %  of  the 
surface  of  a  4-in.  cube  ? 

15.  The  top  of  a  table  equals  300%  of  the  top  of  a 
desk ;  if  the  table  is  3  ft.  square,  what  is  the  area  of  the 
top  of  the  desk  ? 

16.  40  sq.  rd.  equals  what  %  of  an  acre  ? 

17.  A  rectangle  80  rods  wide  and  160  rods  long  equals 
what  %  of  a  square  mile  ? 

18.  i  equals  what  %  of  ^?  of  |  ?  ^  equals  what  %  of  -J-? 
off? 


ADVANCED    ARITHMETIC. 


79 


Draw  rectangles  on  the  blackboard  having  the  ratios  of  the 
units  1,  2,  3,  4,  5,  6,  7,  8. 


1.^  If.  a  equals  100%,  what  does  each  of  the  other  units 
equal  ? 

2.  Add  by  12^%  to  100%. 

3.  Call  a  X  and  name  the  others. 

4.  What  is  the  ratio  of  100%  to  50%?  to  25%  ?  to 
12i%  ?  to  62i%  ?  to  75%  ?  to  87i%  ? 

What  is  the  ratio  of  50%  to  25%?  to  12^%?  to 
62i%?  to87i%? 

5.  12^%  equals  what  part  of  100%?     37^%  of  a  unit 
equals  what  part  of  100%  of  it  ? 

6.  What  is  the  ratio  of  62f%  to  12^%?    What  part  of 
the  unit  is  62|%  of  it  ? 

7.  Pupils  draw  on  blackboard  various  figures  having 
these  ratios. 

Select  figures  in  different  groups  and  have  pupils  give 
ratios  to  other  units. 

8.  If  12^%  is  taken  out  of  a  unit,  what  %  of  the  unit 
remains  ?  ^ 

9.  The  sum  of  75%  and  12-J-%  of  a  unit  equals  what 
%  of  the  unit  ? 

10.  Draw  the  Rectangle  of  which  this  □  equals  37^%  ; 
87i%;  75%. 

11.  I  equals  how  many  thirds  of  |  ?     100%  equals  how 

i"The  knowledge  of  a  color  by  its  name  can  only  be  taught 
through  the  eye.  .  .  .  Technical  terms  must  be  associated  immedi- 
ately with  the  perception  to  which  they  belong."  —  Dr.  Whewell. 


80  ADVANCED    ARITHMETIC. 

many  thirds  of  37^%  ?  |  equals  how  many  thirds  oi' 
37i%  ? 

12.  If  9  pk.  is  37:^%  of  a  unit,  why  may  we  infer  that 
the  unit  equals  |  of  9  pk.  ? 

13.^  Make  similar  sentences:  If  x  equals  621%  of  m, 
^^-  =  m. 

14.  Of  what  unit  is  each  of  the  following  Zl\^o'^  6  pk. ; 
12  pk. ;  15  pk.  ;  270  pk. ;  |  pk. ;  §  pk. ;  |-  pk. 

15.  10  bu.  is  62|%  of  how  many  bu.?  What  part  of 
the  unit  is  given  ?  What  did  you  compare  ?  The  ratio 
of  the  unit  to  62^%  of  the  unit  equals  the  ratio  of  what  to 
10  bu.  ? 

16.  Answer  each  of  the  next  three  questions  and  tell 
what  ratios  are  equal : 

21  yd.  is  871%  of  how  many  yd.? 
$300  is  83i%  of  how  many  dollars  ? 
X  bu.  is  37  J  %  of  how  many  bu.  ? 

17.  Write  sentences  similar  to  the  following;  6  apples 
are  37  J  %  of  8  '2  apples. 

18.  Think  of  a  rectangle  1"  X  5"  and  call  it  5.  Tell  how 
many  %  less  than  5  are  4,  3,  2,  and  1,  respectively. 

19.  Assume  6  equal  objects  to  be  100%.  Tell  how 
many  %  less  than  6  are  5,  4,  3,  2,  and  1,  respectively.  Tell 
how  many  %  more  than  6  are  7,  8,  9,  10,  11,  12,  and  13, 
respectively. 

20.  What  is  12i%  of  640  acres  ?  How  many  sq.  rd.  in 
121%  of  1  acre  ?  320  rd.  equals  what  %  of  1  mi.  ?  How 
many  rd.  in  12J  %  of  1  mi.  ? 

21.  32  equals  what  %  of  64  ?  of  256  ?  of  128  ?  of  48  ? 

22.  63  is  what  %  of  72  ?  of  84  ?  of  126  ? 

1  "  The  growth  of  simple  reasoning  involves  representation  of  per- 
ceptions ;  and  the  growth  of  complex  reasoning  involves  representa- 
tion of  the  results  of  simple  reasoning."  —  Herbert  SpenceB: 


ADVANCED   ARITHMETIC.  81 

23.  63  is  what  %  when  168  is  100%  ? 

24.  Sold  40  sheep  of  a  flock  of  64;  what  %  was  left? 
What  %  was  sold  ? 

What  is  the  ratio  of   the  number  sold  to  the  number 
left? 

25.  Make  similar  sentences :  If  37|%  is  45  ft.,  62^% 
5 -45  ft. 


To  Teacher.  —  Permit  no  attempt  to  solve  problems  which  are 
not  in  the  mind.^  Until  a  question  is  entertained,  it  cannot  be 
answered. 

Through  the  ease  and  correctness  of  the  pupil's  expression  both 
by  hand  and  by  language,  decide  whether  the  work  is  adapted  to 
his  ability.  The  pupil  who  finds  difficulty  in  seeing  62|%  as  |  of 
37|%  is  not  ready  to  make  the  statement  called  for  in  Prob.  25. 
"  Explaining  "  in  such  work  tends  to  clog  the  mind.  The  pupil 
sees  or  he  does  not  see.  The  preparation  for  complex  problems  is 
the  gradual  development  of  more  complex  faculty.  We  promote 
such  development  by  exercise  in  right  directions.  The  average 
pupil  whose  powers  have  been  fitly  exercised  in  earlier  work  sees 
such  relations  with  but  little  effort.  The  act  of  attention  to  the 
particular  aspect  to  which  attention  is  solicited  is  about  all  that  is 
required. 

26.  8/  is  25%  of  what  my  arithmetic  cost;  my  reader 
cost  87:^%  as  much  as  my  arithmetic.  Find  the  cost  of 
the  reader. 

27.  If  a  15-ft.  line  lacks  37^%  of  the  length  required, 
what  is  the  length  required  ? 

28.  The  line  a  equals  12|^%  of  the  line  h.  ^  the  line  a 
equals  what  %  of  -J-  the  line  &  ?  A  line  3  times  as  long  as 
a  equals  what  %  of  the  line  3  times  as  long  as  ^  ? 

29.  The  weight  of  a  1-in.  iron  cube  equals  what  %  of 
tlie  weight  of  a  2-in.  iron  cube? 

1  See  note,  p.  73,  "  Elementary  Arithmetic." 


82 


ADVANCED    ARITHMETIC. 


Place  before  pupils  solids  pictured  on  page  77. 

1.  Call  a  1,  and  name  the  others. 

2.  \  equals  what  %  of  J  ?  of  J  ?  of  f  ?  of  |  ? 

3.  \  equals  what  %  of  |  ?  of  f  ?  of  J  ? 

4.  2  equals  what  %  of  |  ?  of  J  ? 

5.  1  equals  what  %  of  f  ?  of  |  ? 


1.  Observe  the  rectangle. 

What  %  of  J  is  ^  ?  of  J  is  i  ?  of  f  is  ^  ?  of  I  is  §  ?  of  ^ 
is  1?  of  1  is  J  ?  of  i  is  f  ?  of  f  is  f  ?  of  |  is  1  ? 

2.  \  equals  how  many  hundredths  of  J  ?  of  |  ? 

3.  How  many  hundredths  of  J  is  J  ?  of  J  is  J  ?  of  f  is 
J  ?  of  I  is  I  ?  of  1  is  I  ?  of  J  is  J  ?  of  J  is  J  ?  of  J  is  §  ? 
of  f  is  I  ?  of  I  is  1  ? 

4.  I  is  what  %  of  each  of  the  different  numbers  of 
sixths  in  f  ? 

5.  Of  what  is  each  of  the  following  16f  %  ?  2  da.  ; 
4  da.;  3  da.;  $i;  -| ;  $10;  $6;  i;  $.50;  90  bu.;  12  sheep; 
120  sheep;  16|/;  33i  ft. 

6.  A  teacher  bought  60  pencils.  16|%  of  them  were 
lead  and  the  rest  were  slate  pencils.  How  many  slate 
pencils  did  she  buy  ? 

7.  A  man  pays  $20  for  house  rent.  The  rent  equals 
16f  %  of  his  salary.     What  is  his  salary  ? 

8.  A  boy  picks  24  qt.  of  berries,  and  receives  for 
picking  33  J  %  of  the  berries.  How  many  quarts  does  he 
receive  ? 


ADVANCED    ARITHMETIC.  83 

9.  In  a  school  of  200  pupils  J  of  J  are  drawing  and 
the  others  reading.  What  %  of  the  pupils  are  drawing? 
What  reading?  The  number  reading  equals  what  %  of 
the  number  drawing  ? 

10.  I  pay  $3  for  one  book  and  ^^^o  of  $2  for  another. 
How  much  do  both  cost? 

11.  \  equals  33J%  of  what  ?  \  equals  16§%  less  than 
what  part  of  100%  ? 

12.  A  man  invests  |  of  his  money  in  land.  What  % 
does  he  invest  ? 

13.  Show  by  drawing  the  ratio  of  20%  to  ^  of  100%. 

14.  The  area  of  a  rectangle  6"  X  1"  equals  what  %  of  the 
area  of  the  rectangle  3"  X  1"?  What  is  the  ratio  of  f  of 
each  rectangle  to  itself?  to  the  other  rectangle  ?  of  100% 
of  each  rectangle  to  itself  ?  to  the  other  rectangle  ? 

15.  What  %  of  1  is  f  ?  of  f  is  1  ? 

16.  \  of  66f  %  of  the  length  of  a  sidewalk  equals  how 
many  thirds  of  it  ? 

17.  I  buy  a  coat  for  $12  and  sell  it  for  $10.  What 
is  the  ratio  of  the  selling  price  to  the  cost  ?  If  I  sell  at  a 
profit  of  16f  %,  what  shall  I  receive  for  the  coat  ? 

18.  The  rectangle  x  equals  16f  %  of  the  rectangle  y. 
What  is  the  ratio  of  ?/  to  ic  ? 

19.  The  difference  between  the  rectangles  equals  what 
%  of  X  ?  of  7/  ? 

20.  A  boy  made  a  cent  on  every  5  papers  he  sold. 
What  was  his  %  profit  if  the  papers  cost  \^  each  ?  If 
he  sells  60  papers  how  much  does  he  make  ? 

21.  A  merchant  sells  flour  for  $5.50  which  cost  $4.00. 
What  %  is  gained  on  each  barrel  ?  on  100  barrels  ? 

22.  A  field  \  mile  square  equals  what  %  of  a  square 
mile? 

23.  A  man  owned  a  section  of  land.  He  sold  the  S.  E.  \ 
of  the  N.  W.  \.     What  %  did  he  sell  ? 


84 


ADVANCED   ARITHMETIC. 


Draw  a  square  on  the  blackboard  V  long,  a  rectangle  1"  by 
10''  long,  and  a  square  10''  long. 


1.  What  ratios  do  you  see  ? 

2.  If  c  is  100%,  what  isb?     What  is  a  ? 

3.  Show  me  different  parts  of  c  and  tell  me  what  % 
yon  have  shown  me.     Ux.    This  is  -J  of  c  or  50%  of  c. 

4.  1%  of  a  unit  equals  what  part  of  it  ?     1%  of  the 
blackboard  equals  what  part  of  it  ? 

5.  What  is  the  ratio  of  a  to  f  of  a  ?  of  c  to  f  of  c  ?  of 
bto^ofb? 

6.  If  c  is  100%,  wliat  is  ^  of  ^>  ? 

7.  If  cis  1.00,  what  is  b?  a? 

Represent  by  drawing :  .03;  .07;  .05;  .02;  .08;  .37. 

8.  What  %  of  $300  is  a,  if  c  is  $300?     If  c  is  $300, 
what  %  of  $300  is  a  unit  equal  to  7  <x's  ? 

9.  If  cis  100%,  what  is  a?    What  %  is  J  of  a?    Show 
mei%. 

What  is  the  ratio  of  1%  to  i%?    of  10%  to  i%?   of 
100%  toi%? 

10.  What  is  the  ratio  of  1%  to  |%  ?  of  10%  to  |-%  ? 
What  is  the  ratio  of  100%  to  f  %  ? 

Ans.    100  •  f  is  the  ratio  of  100%  to  I  %. 

11.  Give  as  above  the  ratio  of  100%  to  §%?    to  f%  ? 


to|% 


toi%? 


toi%?  toi%? 


ADVANCED   ARITHMETIC.  85 

1.  f  is  the  ratio  of  what  to  12  ?  What  is  the  ratio  of 
9  to  12  ?     I  of  12  equals  what  ?     -f  of  9  equals  what  ? 

What  is  the  ratio  of  the  sum  of  9  and  12  to  12  ?  to  9  ? 
5  is  the  ratio  of  what  to  9  ?  f  is  the  ratio  of  9  to  what  ? 
What  is  the  ratio  of  the  difference  between  9  and  12  to 
each  ?     What  is  the  ratio  of  each  to  the  difference  ? 

2.  If  f  is  the  ratio  of  x  to  y,  what  is  the  ratio  of  y  to 
X  ?  What  is  the  ratio  of  their  sum  to  3/  ?  of  their  sum 
to  oj  ?  What  is  the  ratio  of  y  to  the  sum  ?  of  ic  to  the 
sum  ?  What  is  tlie  ratio  of  their  difference  to  ?/  ?  to  a;  ? 
What  is  the  ratio  of  y  to  their  difference  ? 

3.  If  75%  is  the  ratio  of  a  to  6,  what  is  the  ratio  of  h 
to  a  ?  Wliat  is  the  ratio  of  their  sum  to  a  ?  to  6  ?  What 
is  the  ratio  of  a  to  the  sum  ?  to  ^  ?  of  6  to  the  sum  ?  What 
is  the  ratio  of  their  difference  to  a  ?  to  6  ?  of  a  to  the  dif- 
ference ?  of  Z»  to  the  difference  ? 

4.  Divide  the  blackboard  so  that  50%  of  one  part  equals 
33^%  of  the  other  ;  66f  %  of  the  other.  What  is  the  ratio 
of  the  blackboard  to  each  part  ? 

5.  If  50%  of  the  rectangle  a  equals  SS^^o  of  the  rect- 
angle b,  what  is  the  ratio  of  <^  to  ^  ? 

6.  66f  %  of  the  line  a  equals  75%  of  the  line  b.  What 
is  the  ratio  of  a  to  b? 

7.  66f  %  equals  the  ralio  of  a  to  b.  What  equals  the 
ratio  of  :^  of  a  to  ^  of  ^  ? 

What  %  of  a  added  to  a  equals  b  ?  What  %  of  ^  should 
be  added  to  a  to  make  the  sum  equal  to  b?  The  sum  of 
a  and  b  equajs  how  many  sixths  of  b?  What  %  of  Z>  ? 
of  «^? 

8.  Draw  a  line  and  call  it  10  yd.  Divide  it  into  two 
parts  so  that  the  relative  length  of  the  parts  shall  equal 
that  of  50%  of  a  yd.  to  75%  of  a  yd.  What  is  the  length 
of  each  part  ? 

9.  Two  boys  pay  24/  for  a  book.     The  amount  one 


86  ADVANCED    ARITHMETIC. 

pays  equals  33^%  of  the  amount  the  other  pays.  What 
does  each  pay  ? 

10.  When  the  time  past  noon  equals  33^%  of  the  time 
to  midnight,  what  is  the  time  ? 

11.^  What  is  the  ratio  of  m  to  66J%  of  ???.  ?  If  66|%  of 
m  equals  k,  m  equals  what  %  of  A^  ? 

12.  d  equals  what  %  of  f  of  cZ  ?  If  f  of  (^  equals  y,  d 
equals  what  %  of  ?/  ? 

13.  The  ratio  of  a  to  c?  is  100  ;  of  c  to  c^  f .  What  is  the 
ratio  of  a  to  c  ? 

14.  Give  problems  similar  to  above. 

15.  12^%  of  X  equals  16|%  of  b.  What  is  the  ratio  of 
xtob? 

16.  If  75%  of  a  equals  50%  of  6,  a  equals  what  %  of  ^>? 
Give  similar  problems. 


r     " 
I 


m  n 

1.  If  m  is  100%,  what  is  m  —  7i?  If  m  is  100%,  n 
equals  what  %  ? 

2.  I  is  the  ratio  of  a  to  ^.  Draw  a  and  6.  a  is  what 
%  more  than  6  ?     ^  is  what  %  less  than  a  ? 

3.  I  have  25%  more  money  than  you.  My  money 
equals  what  %  of  yours?     Yours  equals  what  %  of  mine? 

Represent  by  rectangles  the  conditions  showing  ratios  sought 
in  problem  4. 

4.  Gained  20%  by  a  sale  of  goods.  What  was  the 
ratio  of  the  cost  to  the  selling  price  ?  of  the  selling  price 
to  the  cost? 

1  In  none  of  this  work  permit  pupils  to  become  lost  in  a  maze  of 
words.  To  say  "since"  and  "therefore"  is  no  evidence  of  mental 
action.  Turn  to  the  concrete  whenever  there  is  hesitation,  but  see 
that  there  is  sufficient  variety  in  things  used  to  keep  the  mind  active. 


ADVANCED    ARITHMETIC.  87 

5.  If  X  equals  \  more  than  ?/,  what  part  of  x  equals  %j  ? 
What  is  the  ratio  of  ic  to  ?/?  of  v/  to  a?  ? 

Draw  figures  having  these  ratios  : 

6.  If  the  rectangle  k  equals  \  more  than  the  rectangle 
A,  what  part  of  the  rectangle  k  equals  the  rectangle  A? 
The  difference  between  the  two  rectangles  equals  what 
part  of  ^  ?    what  part  of  A  ?    what   %  of  each  ? 

7.  Charles  drew  a  line  and  then  erased  33  J  %  of  it. 
The  length  of  the  remaining  line  was  20  inches.  What 
was  the  length  of  the  line  first  drawn  ? 

8.  What  number  diminished  by  a  sum  equal  to  33^% 
of  itself  equals  20  ? 

9.  9  minutes  is  25%  less  than  the  time  required  to 
sweep  room  G.  What  is  the  time  required  to  sweep  room 
G? 

10.  If  m  equals  .3  more  than  n,  what  is  the  ratio  of  m 
to  71  ?  of  71  to  m  ? 

11.  If  ^  is  12^%  less  than  a,  what  is  the  ratio  of  a  to 
d^  oi  d  to  a ?  of  a  to  the  sum  of  a  and  cZ ?  of  the  sum  to 
a?  to  ^? 

Change  the  next  seven  problems,  thus  : 

12.  18  equals  \  more  than  what  unit  ? 
Change  to,  What  is  the  ratio  of  the  unit  to  18  ? 

13.  36  equals  f  more  than  what  unit? 

14.  21  equals  \  less  than  what  unit  ? 

15.  192  ft.  equals  f  less  than  what  unit  ? 

16.  24  equals  .3  more  than  what  unit  ? 

17.  $49  equals  .3  less  than  how  many  dollars? 

18.  2420  equals  %  more  than  what  unit  ? 

19.  2420  equals  %  less  than  what  unit  ? 

20.  Answer  questions  from  12  to  19,  thus  :  18  equals  \ 
more  than  15. 


88  ADVANCED    ARITHMETIC. 

Change  the  form  of  the  following  questions,  thus  : 

1.  If  20%  be  added  to  Henry's  age,  the  sum  will  be 
24  yr.     How  old  is  he  ? 

Change  to,  If  24  yr.  equal  120%  of  Henry's  age,  how 
old  is  he? 

What  is  the  ratio  of  Henry's  age  to  24  years  ?  What 
is  his  age? 

2.  A  coal  dealer  in  selling  coal  at  f  9  a  ton  received 
20%  more  than  it  cost  liim.  What  did  it  cost  him?  What 
is  the  ratio  of  100%  to  120%  ? 

3.  A  merchant  sold  cloth  at  an  advance  of  33^%  on 
the  cost,  receiving  ^1  per  yd.  What  was  the  cost 
per  yd.  ? 

4.  A  grocer  sold  sugar  at  4/  per  lb.  and  lost  20%  of 
the  cost.     What  did  it  cost  ? 

5.  A  mason  built  27  ft.  of  wall,  which  was  25%  less 
than  his  contract  specified.  How  much  wall  did  he  con- 
tract to  build  ? 

-      6.    42  ft.  is  27%  less  than  the  height  of  a  tree.     What 
is  its  height  ? 

7.  The  number  of  pupils  in  daily  attendance  at  a  school 
is  370,  which  is  17%  less  than  the  number  enrolled.  How 
many  pupils  are  enrolled  ? 

8.  A  boy  spent  40%  of  his  money  for  a  ball  and  Ih'^o 
for  marbles  and  had  70/  left.  How  much  money  had  he 
at  first? 

9.  If  £c,  less  the  sum  of  20  %  and  70%  of  x,  equals  m, 
what  is  the  ratio  of  x  to  m? 

10.  Sold  a  horse  for  $180,  and  gained  \  of  its  cost. 
What  was  its  cost? 

11.  A  table  was  sold  for  f  18,  which  was  \  less  than  its 
value.     What  was  its  value  ? 

12.  A  man  sold  his  house  for  $2500  and  lost  12^%. 
What  did  the  house  cost? 


ADVANCED    ARITHMETIC.  89 

13.  State  the  equality  of  ratios  in  each  problem  from  1 
to  12. 

Ex.  The  ratio  of  100%  to  87 1%  equals  the  ratio  of  the 
cost  of  the  house  to  $2500. 

14.  7)1  equals  the  gain  when  berries  are  sold  at  a  profit 
of  20%.  What  is  the  ratio  of  the  cost  to  m  ?  to  the  sell- 
ing price  ? 

1.  I  paid  $80  for  a  buggy  and  sold  it  for  16|%  more 
than  it  cost.     How  much  did  I  receive  for  it  ? 

2.  The  bread  made  from  a  barrel  of  flour  weighs  37 J  % 
more  than  the  flour.     What  is  the  weight  of  the  bread  ? 

The  weight  of  the  bread  equals  what  %  of  the  weight  of 
the  flour  ? 

The  weight  of  the  bread  equals  how  many  eighths  of  the 
weight  of  the  flour  ? 

3.  Of  what  are  15,  7 J,  9i,  72  and  18,  respectively,  25%  ? 
Of  what  are  13,  J,  f,  2\,  respectively,  16|%  ? 

4.  A  fruit  dealer  sold  pears  at  a  profit  of  $2  a  barrel, 
which  was  a  gain  of  20%.     What  did  they  cost  ? 

5.  If  a  miller  takes  4  qt.  for  every  bu.  of  grain  he 
grinds,  what  %  does  he  take  ? 

6.  In  a  bag  of  27  marbles,  6  are  white.  What  %  of  the 
marbles  are  dark  ? 

7.  If  for  the  use  of  $200  for  1  yr.  a  man  pays  a  sum 
equal  to  .05  of  $200,  what  does  he  pay  at  this  rate  for  the 
use  of  $200  for  4  yr.  ? 

8.  At  6%  per  annum  what  part  of  $80  equals  the  in- 
terest of  $80  for  ^  yr.  8  mo.  ? 

9.  $4  is  paid  for  the  use  of  $50  for  1  yr.  What  is  the 
rate  %  ?  What  is  the  ratio  of  $4  to  $50  ?  Express  in 
hundredths. 

10.  A  block  2"X2"X1"  equals  what  %  of  a  block 
2"X3"X4"? 


90  ADVANCED    ARITHMETIC. 

1.  A  lad  collected  $200  worth  of  bills  at  6%  commis- 
sion.    How  much  was  his  commission  ? 

2.  A  broker  sold  a  lot  for  $400  and  received  a  4J% 
commission.     How  much  was  his  commission  ? 

3.  If  I  pay  an  agent  3%  commission,  what  is  the  cost 
of  every  dollar's  worth  of  goods  bought  ?     Why  ? 

4.  A  lawyer  received  $50  for  collecting  a  note  of  $500. 
What  was  his  rate  of  commission  ? 

5.  A  broker  sold  a  farm  for  $6000  and  charged  $300 
commission.     What  was  the  rate  of  commission  ? 

6.  A  store  worth  $3000  was  insured  for  an  amount 
equal  to  .01  of  its  value.     How  much  was  the  premium  ? 

7.  What  is  the  premium  for  insuring  a  house  worth 
$500  at  3%  ? 

8.  What  must  be  paid  for  insuring  a  building  valued  at 
$3000,  for  §  of  its  value,  at  2J%  ? 

9.  The  premium  for  insuring  a  library  worth  $300  was 
$6.     What  was  the  rate  ? 

10.    When  the  premium  is  $6  and  the  rate  3%,  what  is 
the  value  of  the  thing  insured  ? 

1.  A  merchant  imported  200  yd.  of  silk,  invoiced  at 
$1.50  per  yd.     How  much  was  the  duty  at  20<^  ? 

2.  A  man  has  real  estate  worth  $1000  on  which  he  is 
taxed  lj%.     How  much  is  his  tax  ? 

3.  If  the  rate  is  5  mills  on  a  dollar,  and  the  tax  $70, 
what  is  the  amount  taxed  ? 

4.  If  a  man's  property  is  assessed  at  $14,000,  and  he 
pays  $70  tax,  what  is  the  rate  ? 

5.  A  man  bought  butter  at  15/  a  lb.  and  sold  it  for  25/ 
a  lb.  What  %  did  he  gain  ?  If  he  bought  it  at  25/  a  lb. 
and  sold  it  at  15/  a  lb.,  what  %  did  he  lose  ? 

6.  12i  %  of  96  equals  83J%  of  what  number? 

7.  What  number  equals  f  of  20%  of  80  ? 


ADVANCED    ARITHMETIC.  91 

8.  66^%  of   ^120  equals  |  of   what  was    paid  for  a 
watch.     How  much  did  the  watch  cost  ? 

9.  A  watch  bought  for  $120  was  sold  for  $90.     What 
%  was  lost  ? 

10.  Make  statements  similar  to  the  following:  By  sell- 
ing gingham  for  60/,  I  gain  25%.  The  ratio  of  the  selling 
price  to  the  cost  is  |.     The  cost  is  48/. 

Wall  paper  selling  for  16/  per  roll  brings  33  J  %  profit. 
The  ratio  of  the  selling  price  to  the  cost  is  J ;  of  the 
cost  to  the  selling  price  is  |.     The  cost  is  12/. 

11.  Sold  pine  slabs  for  $3  and  lost  16§%.  The  selling 
price  equaled  how  many  sixths  of  the  cost  ?  What  was 
the  cost  ? 

12.  Make  statements  similar  to  the  following :  Sold 
paints  for  $1125  and  gained  25%.  |  is  the  ratio  of  $1125 
to  the  cost.     The  cost  was  f  of  $1125,  or  $900. 

13.  Make  problems  about  %  of  gain  or  loss  in  buying 
and  selling  coffee,  sugar,  butter,  tea,  apples,  oranges,  milk, 
ice,  chinaware,  table  linen. 

14.  Pineapples  costing  30/  were  sold  at  20%  profit. 
For  how  much  were  they  sold  ? 

15.  A  book  bought  at  $1.25  was  sold  at  a  profit  of  30%. 
For  how  much  was  it  sold  ? 

16.  If  20%  is  the  gain  on  peaches  sold  at  $2.40,  what  is 
the  cost  ?  If  selling  at  $2.40  is  selling  at  a  loss  of  25%, 
what  is  the  cost  ? 

17.  There  is  a  loss  of  37^%  on  damaged  silk  sold  at 
$1.25  a  yd.  What  is  the  cost  ?  What  is  the  loss  on  40 
yd.  ?     What  is'^the  %  of  loss  on  40  yd.  ? 

18.  When  the  ratio  of  loss  to  the  cost  of  goods  is 
12^%,  the  selling  price  equals  what  per  cent  of  the  cost? 
Wliat  is  the  ratio  of  the  cost  to  the  selling  price  ? 

19.  A  boy  sold  a  bicycle  for  $30,  which  equaled  80%  of 
the  cost.     What  was  the  cost  ? 


92  ADVANCED   ARITHMETIC. 

1.  A  man  deposited  $200  in  a  bank  and  each  year  in- 
creased his  deposit  100%.  How  much  was  his  deposit  at 
the  end  of  3  years  ? 

2.  Tea  that  cost  $1.20  was  sold  at  a  profit  of  12i%. 
What  was  the  selling  price  ?  What  is  the  ratio  of  the 
selling  price  to  $1.20  ? 

3.  At  how  much  per  yd.  must  cloth  that  costs  20/  a 
yd.  be  sold  to  gain  20%  ?  cloth  that  costs  25/  a  yd.  ?  that 
costs  35/  a  yd.  ? 

4.  What  will  eight  $100  shares  of  telegraph  stock  cost 
at  4%  premium  ?  at  4%  discount? 

5.  A  man  bought  10  shares  of  nursery  stock  at  par  and 
sold  them  at  a  premium  of  6%.     What  was  his  profit? 

6.  A  dividend  of  8%  is  declared.  How  much  does  a 
stockholder  who  owns  three  $100  shares  receive  ? 

7.  The  ratio  of  the  rectangle  a  to  the  rectangle  b  is  §.  The 
difference  between  the  rectangles  equals  what  %  of  />  ?  of  a? 

If  the  sum  of  the  areas  of  the  two  rectangles  is  22  sq.  in., 
what  is  the  area  of  each  rectangle  ? 

8.  A  merchant  sold  goods  for  a  sum  equal  to  |J  of  the 
cost  and  gained  $4.40.     What  was  the  cost  ? 

9.  A  grocer  sold  nuts  at  a  profit  of  16  §%  and  gained 
42/.     What  was  tlieir  cost  ? 

10.  A  stationer  buys  paper  at  $1  a  ream  and  sells  it  at 
i/  a  sheet.     Does  he  gain  or  lose  ?     What  %  ? 

11.  If  J  acre  is  sold  for  what  an  acre  cost,  what  %  is 
gained  ? 

12.  If  J  of  a  quantity  is  sold  for  a  sum  equal  to  the 
cost  of  j  of  it,  what  %  is  gained  ? 

13.  If  I  of  a  quantity  is  sold  for  a  sum  equal  to  i  the 
cost  of  the  whole,  what  %  is  lost  ? 

14.  §  of  the  sum  received  for  an  article  equals  &  of  what 
was  paid  for  it.  What  is  the  gain  %  ?  What  is  the  ratio 
of  the  sum  received  to  the  cost  ?     Show  by  drawing. 


ADVANCED    ARITHMETIC.  93 

15.  By  a  mistake  in  weighing,  an  amount  of  coal  so  much 
less  than  a  ton  was  sold  for  a  ton  that  the  seller  gained  8%. 
What  part  of  a  ton  was  sold  for  a  ton  ?  What  %  did  the 
buyer  lose  ?     Show  conditions  by  drawing. 

16.  Groods  that  cost  ic/  a  yd.  are  marked  to  sell  at  25% 
profit,  but  are  sold  at  25%  less  than  marked  price.  Draw 
rectangles  showing  ratio  of  marked  price  to  cost  price ;  of 
selling  price  to  marked  price ;  to  cost  price. 

17.  Make  many  statements  similar  to  the  following: 
Goods  marked  to  sell  at  331%  profit  were  sold  at  50% 
below  marked  price.  The  ratio  of  the  selling  price  to  the 
cost  is  §. 

18.  To  sell  at  cost,  goods  marked  at  25%  profit  should 
be  sold  for  what  %  below  marked  price?  goods  marked 
at  331%  profit?  20%  ?  12^%  ? 

19.  If  goods  are  marked  at  50%  profit,  at  how  much 
below  marked  price  must  they  be  sold  to  gain  10%  ?  10% 
below  marked  price  at  25%  profit  equals  what  part  of  the 
cost  ?     What  is  the  ratio  of  40%  to  150%  ? 

20.  A  dealer  lost  8%  of  a  box  of  fruit.  At  what  % 
above  cost  must  he  sell  that  he  may  lose  nothing  ?  that  he 
may  gain  12%  ? 

By  no  manipulation  of  figures  and  formulas  can  the  pupil  arrive 
at  truth  concerning  things.  Sensible  experience  is  in  all  cases 
the  basis.  Gauss  called  geometry  "  a  science  of  the  eye,"  as 
Thiersch  had  before  called  work  in  elementary  mathematics  a 
"thinking  with  the  eye." 

1.  What  is  the  ratio  of  110%  to  100%  ?  How  much 
currency  can  be  obtained  for  $50  in  gold  when  gold  is  at  a 
premium  of  10%  ?  100%  ?  150%  ? 

2.  1.06  is  the  ratio  of  what  to  100%  ?  When  stock  is 
at  6%  premium,  what  is  the  market  value  of  $1  ?  of  $100? 
of  $700? 


94  ADVANCED    ARITHMETIC. 

3.  When  stock  is  at  16§  %  discount,  what  is  the  market 
value  of  stock  whose  par  value  is  $40?  $100?  $1000? 
What  ratios  are  equal? 

4.  Sold  muslin  at  45/  a  yd.  and  gained  12^%.  How 
much  did  it  cost  ? 

5.  Find  the  cost  of  coal  sold  at  $7,  the  loss  being  12^%. 

6.  25%  of  800  bu.  equals  12^%  of  how  many  bushels  ? 

7.  A  man  sold  goods  at  a  gain  of  15%.  His  profit  was 
$60.     For  how  much  did  he  sell  them  ? 

8.  What  is  the  amount  of  sales  when  the  commission 
atli%  is  $300? 

9.  A  man  sold  a  house  at  a  profit  of  $360,  which  was 
6%  more  than  it  cost  him.  How  much  did  he  pay  for  the 
house?     For  how  much  did  he  sell  it? 

10.  What  will  10  shares  of  stock  cost  at  10%  below 
par,  if  I  pay  a  broker  |^%  for  buying,  or  brokerage  ? 

11.  What  is  the  annual  income  on  a  bond  of  $4000  which 
yields  6  %  annually  ? 

1.  What  is  the  interest  on  $70  for  1  yr.  at  8%?  for 
2  yr.  ?  for  1^  yr.  ?  for  6  mo.  ? 

2.  When  $300  is  loaned  at  7%  per  yr.,  what  is  the 
amount  of  the  principal  and  interest  in  1^  yr.  ? 

3.  When  the  interest  on  $100  for  1  yr.  is  $7,  what  is 
the  rate  ? 

4.  When  the  interest  on  $100  for  2  yr.  is  $16,  what  is 
the  rate  ? 

5.  What  principal  at  6%  yields  $300  in  1  yr.? 

6.  What  principal  yields  $350  in  3^  yr.  at  10%? 
What  %  does  any  principal  earn  in  3^  yr.  at  10%? 

7.  At  what  %  does  $75  in  2  yr.  amount  to  $91  ?  How 
much  interest  does  the  $75  yield  in  2  yr.  ? 

8.  At  what  %  per  annum  does  $50  amount  to  $65  in 


ADVANCED    ARITHMETIC.  95 

9.  At  what  rate  does  $500  gain  $50  in  1  yr.  ?  in  2  yr.  ? 
inliyr.? 

10.  If  $800  yields  $120  in  2^  yr.,  what  does  it  yield  in 
1  yr.  ?     What  is  the  rate  ? 

11.  At  what  %  does  $900  in  3  yr.  yield  $180  interest  ? 

12.  At  what  rate  per  annum  does  $100  gain  $25  in 
4^  yr.  ?     What  part  of  the  $25  is  gained  in  1  yr.  ? 

13.  At  what  %  per  annum  does  a  principal  double  itself 
in  4  yr. ?  in  5  yr. ?  in  9  yr.? 

14.  At  what  rate  must  $320  be  loaned  to  yield  $16  in 
lyr.? 

1.  If  5%  is  the  ratio  of  h  to  k,  what  is  the  ratio  of  k 
to  h?  of  their  sum  to  k?  of  their  sum  to  A?  of  their 
difference  to  k?  of  their  difference  to  h?  of  k  to  their 
difference  ?   of  h  to  their  difference  ? 

2.  .08  is  the  ratio  of  what  unit  to  75  lb.  ?  What  is  the 
ratio  of  6  lb.  to  75  lb.  ?  If  .08  is  the  ratio  of  6  lb.  to  75 
lb.,  what  is  the  ratio  of  75  lb.  to  6  lb.?  of  their  sum  to 
75  lb.  ?  of  their  sum  to  6  lb.  ?  of  their  difference  to  75  lb.  ? 
of  their  difference  to  6  lb.  ? 

3.  What  equals  35%  of  640  acres  ? 
7   32 

xn 

.'.  224  acres  equals  35%  of  640  acres. 

4.  224  acres  equals  35%  of  what  ? 
20  32 

t 

.'.  224  acres  equals  35%  of  640  acres. 


96  ADVANCED    ARITHMETIC. 

5.  224  acres  equals  what  %  of  640  acres  ? 
7 

$$ '  640  acres  equal  what  %  of  640  acres  ? 

n^  ■  100  _  What  is  the   ratio   of   224   acres   to  640 

040      ~      *     acres?     To  what  part,  then,  of  100%  of 
$0  640  acres  is  224  acres  equal  ? 

20 
.*.  224  acres  equal  35%  of  640  acres. 

6.  35%  is  the  ratio  of  what  to  640  acres  ?  What  is 
the  ratio  of  the  sum  of  640  acres  and  224  acres  to  640 
acres  ?  to  224  acres  ?  What  is  the  ratio  of  640  acres  to 
the  sum  ?  of  224  acres  to  the  sum  ? 

1.    What  is  6%  of  75  bu.  ? 
6   75 
100 


=  9 


To  Teacher.  —  Ask  many  questions  similar  to  the  following  : 
6%  is  the  ratio  of  what  to  75  bu.?  What  is  the  ratio  of  75  bu.  to 
4.5  bu.? 

2.  What  is  25%  of  160  ft.? 

3.  What  is  37^%  of  $845  ? 

4.  What  is  8%  of  647  oz.? 

5.  What  is  f  %  of  $824  ?     (What  is  f  of  1  %  of  $824  ?) 
•^  •  824  _  ^        What  equals  1  %  of  $824  ? 

~  •         What,  then,  equals  f  of  1%  of  824  ? 
What  is  I  %  of  876  ft.? 
What  equals  f  %  of  $214? 
What  equals  |-%  of  64.82  tons  ? 
What  equals  2^%  of  $78? 

What  is  the  ratio  of  2i%  of  $78  to  1%  of 
_  ^        $78  ? 
2-100      ■  What  equals  1  %  of  $78  ? 

What,  then,  equals  f%  of  $78? 


4- 

100 

6. 

7. 

8. 

9. 

5' 

78 

ADVANCED    ARITHMETIC.  97 

10.  What  is  31  %  of  729  days  ? 

11.  What  equals  7^%  of  $856.50? 

12.  What  equals  3|%  of  1600  ft.? 

13.  What  equals  13^-%  of  $1672.34? 

14.  What  equals  87^%  of  647  yd.? 

15.  What  equals  mi%  of  5876? 

16.  What  equals  37^%  of  643? 

17.  Make  10  sentences  like  this  :   '^^  of  yj^  of  745  da. 
equals  3f  %  of  745  da. 

18.  Make  10  sentences  like  this :   |  of  940  ft.  equals 
62i%  of  940  ft. 

1.  What  is  the  ratio  of  6  mo.  to  1  yr.  ?    of  9  mo.  ?   of 

4  mo.  ?  of  8  mo.  ?  of  1  yr.  4  mo.?  of  1  yr.  6  mo.  ?  of  1  yr. 
7  mo.  ?  of  2^  mo.  ?  of  1  yr.  2\  mo.  ? 

2.  If  7%  of  %y  equal  the  interest  of  %y  for  1  yr.,  what 
equals  the  interest  of  %y  for  6  mo.?  for  1  yr.  6  mo.?  for 
1  yr.  8  mo.  ?  for  1  yr.  5  mo.  ? 

3.  If  $84  is  the  interest  of  %y  for  1  yr.  9  mo.,  what  is 
the  interest  of  %y  for  1  yr.  ? 

4.  If  $a;  at  8%  yields  $92  in  1  yr.,  what  does  it  yield 
at  the  same  rate  in  9  mo.  ?    in  1  yr.  8  mo.  ? 

5.  What  part  of  $640  equals  the  interest  for  1  yr.  at 
7%?  What  part,  tlien,  of  ^^^  of  $640  equals  the  interest 
for  1  yr.  8  mo.  at  7%? 

6.  What  is  the  interest  of  $640  for  1  yr.  8  mo.  at  7%  ? 

^^_  of  $640  equals  the  interest  for  what 

5  •  7  •  640      ^  ^     time? 

3  ■  100      ~  *  J  of  t5^  of  $640  equals  the  interest  for 

what  time  ? 

7.  What  is  the  interest  of  $600  for  1  yr.  3  mo.  at  9%  ? 

8.  $850  for  2  yr.  7  mo.  at  10%  ? 

9.  $1270  for  3  yr.  4  mo.  at  8%  ? 
10.    $500  for  7  mo.  at  7%  ? 


98  ADVANCED    ARITHMETIC. 

11.  $1250  for  3  yr.  6  mo.  at  7%  ? 

12.  $200  for  9  mo.  at  7^%  ?    Of  $200  for  1  yr.  at  4i%  ? 

1.  f  is  the  ratio  of  what  to  $25  ?  What  is  the  ratio 
of  $30  to  $25?  of  $25  to  $30?  of  their  sum  to  $30?  of 
their  sum  to  $25  ? 

2.  -L§o  is  the  ratio  of  what  to  15/?  What  is  the  ratio 
of  $5  to  15/?  of  15/  to  $5?  of  their  sum  to  15/? 

3.  If  J-2^0.  ig  the  ratio  of  x  to  y,  what  is  the  ratio  of  y  to 
a?  ?  of  their  sum  \^o  y'i  of  their  sum  to  a?  ?  of  y  to  their 
sum  ?  of  a;  to  their  sum  ? 

4.  168  men  equals  8%  of  how  many  men  ? 

100  •  168      ^  The   ratio  of   100%   to  8%  equals  the 

8  '        ratio  of  how  many  men  to  168  men  ? 

5.  Of  what  unit  is  27.5  bu.  7%  ? 

6.  Of  what  unit  is  73  cd.  16%  ? 

7.  255  equals  30%  of  what  unit? 

8.  180  equals  12^%  of  what  unit  ? 

9.  $220.50  equals  107%  of  what  unit  ? 

10.  $75  equals  104%  of  what  unit  ? 

11.  231  gal.  equals  2^%  of  what? 
100-2-231  _  ^      What  part  of  231  gal.  equals  1  %  ? 

5  ~  *       What,  then,  equals  100%  ? 

12.  $846  equals  1^%  of  what  unit? 

13.  854.37  ft.  equals  5j%  of  what  unit? 

14.  247  yd.  equals  2f  %  of  what  unit  ? 

15.  $675.25  equals  2|%  of  what  unit  ? 

16.  $785.56  equals  7f  %  of  what  unit  ?    , 

17.  68  ft.  equals  f  %  of  what  unit? 

18.  2745  equals  f  %  of  what  ? 

19.  Of  what  unit  is  45   |%  ? 

20.  Of  what  unit  is  144  37^%  ? 

21.  210  equals  87^%  of  what  ? 

22.  80  equals  66|%  of  what  unit  ? 


ADVANCED    ARITHMETIC.  99 

23.  Of  what  unit  is  $745  83J  %  ? 

24.  Write  ten  sentences  similar  to  this :  If  594  equals 
S^^o  of  a-  unit,  100  '  f  of  594  equals  the  unit. 

25.  Write  ten  sentences  similar  to  this  :  If  69  ft.  equals 
66 1%  of  a  unit,  |  of  69  ft.  equals  the  unit. 

1.    Express  the  ratio  of  4  to  6  in  different  terms  ;  of  15 
to  25 ;  of  7  to  10 ;  of  2  to  50. 


2.  What  is  the  ratio  of  a  to  6  expressed  in  hundredths  ? 

b  equals  how  many  hundredths  ot  b? 
What  is  the  ratio  of  a  to  b? 

'—  =  .75.  What,  then,  is    the    ratio  of   a  to  100 

hundredths  of  ^  ? 

3.  b  equals  how  many  hundredths  oi  a? 

„^^  a  equals  how  many  hundredths  of  a  ? 

.      '  ^  What  is  the  ratio  oi  b  to  a? 

— -^=1.331.      What,    then,  is   the    ratio   of   b  to   100 
liundredths  of  a  ? 

4.  What  is  the  ratio  of  9  to  12  expressed  in  hundredths  ? 
9- 1-00^, 

12 

5.  What  is  the  ratio  of  7  to  2  J  expressed  in  hundredths  ? 

^'      =?     What  is  the  ratio  of  7  to  2^? 


.      1.   What  ia  the  ratio  of  2^  to  7  ? 

If  7  equals  100%,  what  %  does 
50  2i  equal  ? 

5/00  _  250  _  If  2^  equals  100%,  what  does  7 

X4     ~    7    ~      ^*      equal? 
7  2i  equals  what  %  of  7? 

.-.2^  equals  35f%  of  7. 


100  ADVANCED    ARITHMETIC. 

State  the  %  relation  of  a  to  6  and  of  ^  to  a : 


2.  150  yd.  450  yd.  42.7  .24 

3.  560  inin.  140  min.  1287  543 

4.  720  bu.  370  bu.  §  f 

5.  75  sheep  300  sheep  |  § 

6.  350  329  $5  $5^ 

7.  8i  12  ^4 

8.  85i  72  •  i  f 

9.  6.4  1.6  }  i 

1.  4  J  equals  what  %  of  9f  ? 

13 -4   100  _  ^      9f  equals  what  %  of  9f  ? 

3-39      "  *       Then  what  equals  the  %  that  1  is  of  9J  ? 
Then  what  equals  the  %  that  4 J  is  of  9J  ? 

2.  5^  equals  what  %  of  4f  ? 

3.  4f  equals  what  %  of  5^? 

4.  19^  equals  what  %  of  25§  ? 

5.  6.7  equals  what  %  of  54? 

1.  A  man  invested  $5280  in  cotton  and  sold  it  at  a 
profit  of  12%.     What  was  his  profit? 

2.  A  grocer  bought  500  bananas.  37^%  of  them  were 
not  marketable.     How  many  did  he  sell  ? 

3.  A  man  collected  40%  of  a  note  of  $675  and  charged 
Oi%  commission.  What  was  his  commission  ?  On  what 
did  he  receive  his  commission  ? 

4.  A  man  owed  $347  on  account  and  settled  it  for  cash 
at  2  %  off.     What  was  the  discount  ? 

5.  A  man  paid  a  tax  of  $73.50,  which  was  3%  of  the 
value  of  his  property.     What  was  its  value  ? 

6.  What  was  the  amount  of  the  sales  when  the  com- 
mission paid  was  $185.60  and  the  rate  3i  %  ? 


ADVANCED    ARITHMETIC.  101 

7.  At  7i%  an  attorney  was  paid  $144.06  for  collecting 
a  note.     What  was  the  face  of  the  note  ? 

8.  An  agent  sells  550  bbl.  of  flour  at  $10  a  bbl.  and 
remits  $5000.     What  is  the  rate  of  commission  ? 

9.  An  army  of  1400  men  went  into  a  battle ;  after  the 
battle  there  were  only  900.     What  was  the  loss  %  ? 

10.  A  man's  expenses  are  $500  a  year;  his  income  is 
$1600  a  year.    What  %  of  his  income  equals  his  expenses  ? 

11.  A  chain  is  14  carats  fine.     What  %  of  it  is  gold  ? 

1.  If  15%  of  a  certain  ore  is  silver,  how  much  silver  is 
there  in  4850  lb.  of  ore  ? 

2.  After  a  deduction  of  8%  from  a  bill  of  $416.28  is 
made,  how  much  is  the  bill  ? 

3.  A  boy  bought  bananas  at  $1.50  a  hundred  and  sold 
them  at  $.03  each.     What  %  did  he  gain  ? 

4.  A  man  bought  secondhand  books  at  $10  a  dozen 
and  sold  them  at  $1.50  each.     What  was  his  %  of  gain  ? 

5.  If  the  income  of  $2000  is  $250,  what  is  the  rate  ? 

6.  After  taking  out  30%  of  the  grain  in  the  bin  there 
remained  40  bu.  3^  pk.  How  much  wheat  was  there  at 
first? 

7.  A  broker  bought  stock  at  3%  discount,  and,  selling 
the  same  at  5%  premium,  gained  $560.  How  many  shares, 
each  worth  $100,  did  he  purchase  ? 

8.  What  sum  must  I  invest  in  stock,  at  par,  paying  an 
annual  dividend  of  5 J  %,  to  realize  an  income  of  $2200 
yearly? 

9.  At  what  rate  must  I  invest  a  trust  fund  of  $30,000 
to  secure  an  annual  income  of  $1000  ?  a  semi-annual  in- 
come of  $500  ? 

10.  A  principal  of  a  school  receives  $198  per  month, 
after  a  deduction  of  $2  for  pension  fund  has  been  made. 
What  %  of  salary  is  deducted  for  pension  fund? 


102  ADVANCED    ARITHMETIC. 

1.  A  man  sold  25%  of  1000  bu.  of  potatoes.  The  re- 
mainder was  16|%  of  his  entire  crop.  How  much  was  his 
crop  ? 

2.  $18  was  spent  in  repairing  a  carriage  which  cost 
$105.  It  was  then  sold  for  $160.  What  was  the  %  of 
profit  ? 

3.  150  bu.  of  apples  were  sold  at  $1.25  a  bu.  What  was 
the  rate  of  gain  if  the  cost  was  80/  a  bu.  ? 

4.  A  merchant  marked  silk  at  $2.75  a  yd.  and  sold  it 
at  10%  below  the  marked  price.  For  how  much  did  he 
sell  it  ? 

5.  Bought  150  bbl.  of  flour  for  $850  and  sold  it  at  a 
loss  of  16%.     What  was  the  selling  price  per  bbl.  ? 

6.  If  $60  is  the  amount  paid  for  insuring  15  horses  at 
4-J-%,  what  is  the  value  of  the  horses  ? 

7.  A  man  who  owned  70%  of  a  store  sold  40%  of  his 
share  for  $7000.  What  was  the  value  of  his  share  ? 
What  was  the  value  of  the  store  ? 

8.  Cotton  was  bought  at  15§/  a  lb.  and  sold  at  18-|^/. 
What  was  the  gain  %  ? 

9.  Carpeting  cost  $1.87^  a  yd.  and  was  sold  for  $2.25 
a  yd.  What  was  the  gain  %  ?  What  amount  must  be 
added  to  $1.87^  that  the  sum  shall  be  $2.25? 

10.  A  real  estate  agent  sold  430  acres  at  $75  an  acre 
and  charged  a  commission  of  3-^%.  How  much  was  his 
commission,  and  how  much  was  paid  to  the  land  owner  ? 

1.  Find  the  cost  of  120  shares  of  N.  Y.  Central  R.  R. 
stock  at  87-J-,  brokerage  ■J%.  What  does  $1  of  the  stock 
cost  ?     What  is  the  cost  of  1  share  ? 

2.  A  broker  bought  40  shares  of  R.  R.  stock  at  92  and 
sold  them  at  105.     How  much  did  he  gain  ? 

3.  A  broker  received  $4.80  for  a  draft  of  $1280.  The 
brokerage  equals  what  %  of  the  draft  ? 


ADVANCED    ARITHMETIC.  103 

4.  A  broker  was  paid  $300  for  buying  $15,600  worth 
of  stock.  The  commission  equaled  what  %  of  the  value 
of  the  stock  ? 

5.  Bought  bonds  at  112  and  sold  them  at  115,  making 
$300.     How  many  bonds  of  $1000  each  did  I  buy? 

6.  If  shoes  marked  at  $3.50  a  pair  are  sold  at  10%  dis- 
count, what  is  the  discount  ?     What  is  the  net  price  ? 

7.  A  man  asked  $125  for  a  buggy,  but  for  cash  took 
10  %  off.     How  much  was  deducted  ? 

8.  A  broker  received  $112.50  for  selling  bonds,  charg- 
ing 1%  brokerage.  ¥or  how  much  were  the  bonds 
sold? 

9.  A  man  sells  goods  at  a  profit  of  16%  and  makes 
$48.     What  is  the  cost  ? 

10.    A  man  bought  resin  at  $2.25  a  bbl.  and  sold  it  at 
$5.30  a  bbl.     What  was  the  gain  %  ? 

1.  Paid  $60  for  insuring  a  house  worth  $1200  for  2  yr. 
What  was  the  annual  rate  ? 

2.  In  a  school  of  59  pupils  the  average  daily  attendance 
was  47.     What  was  the  %  of  attendance  ? 

3.  A  mill  worth  $16,000  is  insured  for  f  of  its  value  at 
li%-     What  is  the  premium  ? 

4.  How  much  must  be  paid  for  insuring  a  consignment 
of  goods  worth  $7840  at  2^%  ? 

5.  I  insured  my  house  for  f  of  its  value  at  lJ-%  and 
paid  $130.     What  was  the  value  of  the  house  ? 

6.  A  house  which  was  insured  for  f  of  its  value  was 
burned.  Its  value  was  $4900.  What  %  did  the  owner 
lose? 

7.  When  a  premium  of  $750  is  paid,  the  rate  being 
1-J%,  what  is  the  amount  insured? 

8.  Paid  $187  to  insure  ^  the  value  of  a  store  at  2f  %. 
What  was  the  value  of  the  store  ? 


104  ADVANCED    ARITHMETIC. 

9.  In  a  school  of  50  pupils  there  were  150  absences  in 
8  weeks.     What  was  the  %  of  attendance  ? 

10.  A  cubic  foot  of  water  weighs  62-J  lb.  and  a  cubic 
foot  of  ice  57-J-  lb.  Ice  is  what  %  lighter  than  an  equal 
bulk  of  water  ? 

1.  An  agent  furnished  a  schoolhouse  for  $2000  and 
received  $40  commission.     What  was  the  rate  %  ? 

2.  The  rate  of  tax  in  a  town  is  lj%.  How  much  is 
Mr.  Smith's  tax  if  his  property  is  valued  at  $10,000  ? 

3.  The  taxable  property  of  a  town  is  $1,505,470.  The 
rate  of  tax  for  school  purposes  is  3^^  mills  on  the  dollar. 
What  is  the  amount  of  school  tax  assessed  ? 

4.  If  the  taxable  property  of  a  town  is  $1,505,470  and 
the  tax  collected  $5269.145,  what  is  the  rate  of  taxation  ? 

5.  What  is  the  cash  value  of  a  bill  of  goods  amounting 
to  $497  at  25%  discount  and  5%  off  for  cash  ? 

6.  What  is  the  cash  value  of  a  bill  of  goods  amounting 
to  $7845  at  16%  discount  and  3%  off  for  cash  ? 

7.  What  is  the  duty  at  20%  ad  valorem  on  140  yd.  of 
goods  valued  at  $2  a  yd.  ? 

8.  What  is  the  duty  at  40%  ad  valorem  on  600  yd.  of 
silk  valued  at  $1.75  per  yd.?  For  how  much  a  yd.  must 
the  importer  sell  the  silk  to  clear  35%  ? 

9.  An  attorney  received  $125  for  collecting  rents  to 
the  amount  of  $2150.     What  was  the  rate  of  commission  ? 

10.  If  it  costs  $93.50  to  insure  a  store  for  f  of  its  value 
at  1|%,  what  is  the  value  of  the  store  ? 

1.  A  man  borrows  money  at  6%  per  annum  and  pays 
$48  interest.  What  more  do  you  need  to  know  in  order  to 
find  the  amount  borrowed  ?  If  $48  is  the  interest  paid  in 
6  mo.,  what  was  the  sum  borrowed  ?  if  $48  is  the  interest 
for  1  yr.  ?  for  2  yr.  ? 


ADVANCED    ARITHMETIC.  105 

2.  If  for  the  use  of  money  for  1  yr.  a  man  pays  a  sum 
equal  to  .07  of  what  he  borrows,  in  2  yr.  he  pays  a  sum 
equal  to  what  ?  in  3  yr.  ?  in  4  yr.  and  4  mo.  ? 

3.  A  sum  equal  to  .05  of  $m  is  a  man's  interest  for 
1  yr.     What  equals  his  interest  for  2  yr.  and  6  mo.  ? 

4.  What  sum  of  money  lent  at  6%  yields  $72  interest 
in  1  yr.  ? 

inn'4k^'^  What  is  the  ratio  of  the  principal 

\^^^^  =  $1200.     to  the  interest  for  1  yr.  ? 
0 

5.  $120  is  the  interest  for  1  yr.  at  10%.  What  equals 
the  principal  ? 

6.  What  sum  of  money  produces  $34  interest  in  2  yr. 
and  6  mo.  at  6%  ? 

7.  What  sum  of  money  produces  $42.35  interest  in 
1  yr.  and  6  mo.  at  7%  ? 

8.  What  principal  earns  $24  in  2  yr.  at  6%  ? 

9.  How  much  money  has  a  man  loaned  if  he  receives 
for  its  use  $12  at  the  end  of  3  yr.,  interest  5%  ? 

10.  At  5%  what  principal  yields  $350  in  6  mo.  ? 

11.  What  sum  of  money  produces  $25  in  7  mo.  at  9%  ? 

12.  How  much  must  I  invest  at  6%  for  1  yr.  and  9  mo. 
to  earn  $900  ? 

To  Teacher.  —  Require  pupils  to  make  problems  similar  to  those 
given.  Select  problems  and  have  pupils  use  them  as  a  basis  for 
new  problems.  Ex.  A  broker  sold  property  valued  at  .|112.50, 
at  a  commission  of  3%.     Find  his  commission. 

Make  a  new  problem,  thus  :  What  is  the  interest  on  $112.50  at 
3%?  Or,  How  much  must  be  paid  for  insuring  clothing  worth 
1112.50  at  3%?  Unless  pupils  can  construct  problems  similar  to 
those  given,  and  readily  make  new  questions  by  changing  the  con- 
ditions, their  work  is  mechanical  and  not  mental.  There  should 
be  growing  power  to  discriminate,  to  see  likeness  amidst  diversity, 
to  separate  the  essential  from  the  accidental. 


106  ADVANCED    ARITHMETIC. 

1.  An  auctioneer  sold  $374  worth  of  furniture  and 
charged  a  commission  of  5%.     How  much  did  he  receive  ? 

2.  At  5%  commission  an  auctioneer  received  $18.70 
for  selling  furniture.  For  how  much  was  the  furniture 
sold? 

3.  $18.70  was  the  commission  paid  for  selling  $374 
worth  of  furniture.     What  was  the  rate  of  commission  ? 

4.  Mr.  Smith  bouglit  a  house  for  $4850  and  paid  15%. 
How  much  did  he  still  owe  ? 

5.  A  merchant  sells  $250  worth  of  goods  and  gains 
$50.     What  is  his  rate  of  profit  ? 

6.  What  is  the  interest  on  $275  for  1  yr.  at  6%  ? 

1.  J  is  the  ratio  of  x  to  y.  The  sum  of  x  and  y  is  30. 
Wliat  is  the  value  of  each  ? 

2.  A  boy  received  30/  with  which  to  buy  marbles  after 
taking  out  for  making  the  purchase  \  as  much  as  he 
invested.  How  much  did  he  expend  for  marbles  ?  Show 
by  drawing. 

The  sum  paid  for  the  marbles  equals  what  part  of  30/  ? 
Why? 

3.  The  line  ^ represents  the  amount  an  agent 

receives  with  which  to  make  an  investment  after  deduct- 
ing his  commission  of  20%  ;  show  the  part  of  the  line 
representing  the  investment. 

4.  The  amount  received  equals  the  sum  of  the  amount 
invested  and  what  ? 

"  Reasoning  and  classification  are  the  necessary  comj)lements  of 
each  other.  ...  It  follows  that,  contemplated  from  this  point  of 
view,  reasoning  is  a  classification  of  relations.  But  what  does 
classification  mean?  It  means  the  grouping  together  of  those 
which  are  like  —  the  separation  of  the  like  from  the  unlike.  .  .  . 
The  idea  underlying  all  classification  is  that  of  similarity."  — 
Herbert  Spencer. 


ADVANCED    ARITHMETIC.  107 

5.  The  amount  an  agent  receives  equals  f  of  the 
amount  invested.  The  amount  invested  equals  what  part 
of  the  amount  received  ? 

6.  An  agent  receives  U§  of  the  amount  he  invests. 
What  is  the  ratio  of  the  amount  invested  to  the  amount 
received  ? 

7.  A  man  received  $84  with  which  to  buy  cotton  after 
deducting  for  making  the  purchase  -^^  as  much  as  he 
invested.     What  did  he  pay  for  the  cotton  ? 

The  sum  paid  for  the  cotton  equals  what  part  of  the 
money  received  ? 

8.  An  agent  received  $500  with  which  to  buy  wheat, 
after  deducting  for  making  the  purchase  .05  as  much  as  he 
invested.  What  did  he  pay  for  the  wheat  ?  What  is  the 
ratio  of  the  amount  the  agent  received  to  the  amount 
invested  in  wheat  ? 

If  the  agent  received  .05  of  the  $500,  on  what  would  he 
receive  commission  besides  the  money  invested  in  wheat  ? 

9.  An  agent  received  $250  with  which  to  buy  oranges 
after  deducting  his  commission  at  7%.  How  much  did  he 
expend  for  oranges  ? 

10.  A  Chicago  merchant  sent  his  agent  in  New  York 
$5275.20  to  be  invested  in  coffee  after  deducting  his  com- 
mission at  lj%.  How  much  did  he  expend  for  coffee  and 
what  was  his  commission  ?  What  is  the  ratio  of  the  money 
invested  in  coffee  to  the  amount  the  agent  received  ? 

11.  An  agent  bought  goods  at  f  %  commission  ;  paid  $75 
for  expenses,  smd  sent  a  bill  for  $2847.50.  What  was  the 
amount  of  the  purchase  ? 

12.  A  stock  of  prints  was  sold  at  a  commission  of  1}%, 
and  the  proceeds  invested  in  cambrics  was  $35.  For 
how  much  did  the  prints  sell  ? 

$35  equals  what  part  of  the  selling  price  of  the 
prints  ? 


108  ADVANCED    ARITHMETIC. 

13.  An  agent  received  j\  of  the  selling  price  of  a  horse. 
The  owner  received  $90.  For  how  much  was  the  horse 
sold? 

14.  For  how  much  must  a  lot  be  sold  that  the  owner 
may  receive  $600  after  paying  a  real  estate  dealer  2%  for 
selling  it  ?  The  selling  price  equals  how  many  ninety- 
eighths  of  $600? 

15.  For  how  much  must  a  farm  be  sold  that  the  owner 
may  receive  $5000  and  an  agent  5%  for  selling  it  ?  What 
%  of  the  selling  price  does  the  owner  receive  ? 

What  is  the  ratio  of  the  selling  price  to  $5000  ? 

16.  A  man  wishes  an  agent  to  sell  his  house,  which  cost 
$5250,  for  enough  to  cover  both  the  cost  of  the  house  and 
the  agent's  commission  at  2%.  For  how  much  must  the 
house  be  sold  ? 

$5250  equals  how  many  hundredths  of  the  selling  price  ? 

17.  For  how  much  must  lumber  worth  $27,845  be  in- 
sured at  3%  to  cover  both  the  value  of  the  lumber  and  the 
premium  ? 

18.  For  how  much  must  you  give  a  note  at  a  bank  to 
obtain  $180  if  the  banker  retains  yL  of  the  value  of  the 
note  for  the  use  of  the  money  ? 

19.  If  $250  is  the  net  proceeds  of  a  note  of  which  2% 
was  retained  for  the  use  of  the  money,  what  is  the  face  of 
the  note  ? 

20.  The  amount  paid  the  teachers  in  a  district  is  $3740  ; 
what  amount  of  tax  must  be  assessed  for  the  teachers'  fund 
if  the  cost  of  collecting  the  tax  is  2%  ? 

21.  The  net  proceeds  of  a  tax  assessment,  after  deduct- 
ing 2i%  for  collection,  was  $8794.75,  and  5%  of  the  tax 
was  not  collected.     What  was  the  assessment  ? 

22.  If  you  buy  potatoes  at  45/  a  bu.  and  J  of  them 
spoil,  at  what  price  must  you  sell  the  remainder  that  you 
may  lose  nothing  ? 


ADVANCED    ARITHMETIC.  109 

What  part  of  a  bu.  must  be  sold  for  45/  that  there  may 
be  no  loss  ? 

For  how  much,  then,  must  a  bu.  be  sold  ? 

23.  A  dealer  bought  wool  at  28/  per  lb.  If  in  cleansing 
it  loses  ^,  at  how  much  per  lb.  must  he  sell  the  clean  wool 
to  gain  I  on  the  cost  ? 

24.  A  fruit  dealer  lost  25%  of  a  quantity  of  apples  and 
sold  the  remainder  at  a  gain  of  33J%.  Eequired  the  %  of 
gain  or  loss. 

25.  How  many  yd.  of  ducking,  |  of  a  yd.  wide,  are  re- 
quired to  line  the  carpet  of  a  room  12  ft.  by  15  ft.,  if  the 
ducking  shrinks  4%  ? 

26.  Mr.  Brown  paid  $x  for  a  horse  and  sold  it  for  20% 
more  than  he  paid  and  i  less  than  he  asked  for  it. 

Represent  cost,  selling  price,  and  asking  price. 

27.  A  man  bought  a  horse  for  $72  and  sold  it  for  J  more 
than  it  cost  and  ^^  less  than  he  asked  for  it.  How  much 
did  he  ask  for  it  ? 

28.  Find  the  marking  price  of  goods  that  cost  $15  so  as 
to  reduce  the  marked  price  J  and  yet  make  a  profit  of  i. 

29.  Find  the  marking  price  of  goods  that  cost  $60  so  as 
to  sell  10%  below  the  marked  price  and  still  gain  20%. 

30.  A  man  sold  a  buggy  for  20%  less  than  he  asked  for 
it  and  received  $95,  which  was  20%  more  than  it  cost. 
What  was  the  cost  and  what  was  the  selling  price  ? 

31.  An  agent  receives  a  discount  of  50%  from  the  retail 
price  of  articles  and  sells  them  at  the  retail  price.  What 
is  his  gain  ?       ' 

32.  What  must  I  ask  for  a  horse  in  order  to  fall  33^% 
and  still  make  33i%  ? 

33.  A  merchant  asked  25%  more  for  goods  than  they 
cost  him,  but  at  last  sold  them  at  a  reduction  of  25%  from 
his  asking  price.  If  the  cost  of  the  goods  whie  $150,  did 
he  gain  or  lose,  and  what  %  ? 


110  ADVANCED   ARITHMETIC. 

34.  $3745  less  the  agent's  commission  at  3%  equals 
the  sum  Mr.  A  invested  in  wheat.  How  much  was  the 
agent's  commission  ? 

35.  At  2%  commission  an  agent  received  $282  for  the 
purchase  of  apples  at  $3  a  bbl.  How  many  bbl.  of  apples 
did  he  purchase  ? 

36.  The  cost  of  a  horse  and  saddle  is  $180 ;  20%  of  the 
cost  of  the  horse  equals  the  cost  of  the  saddle.  What  is 
the  cost  of  each  ? 

37.  20%  of  a  pole  is  in  the  mud,  33  J  %  in  the  water, 
and  14  ft.  in  the  air.  How  long  is  the  part  in  the  mud 
and  water  ? 

38.  A  man  gave  $150  for  a  watch  and  chain.  The  cost 
of  the  chain  was  equal  to  25%  of  the  cost  of  the  watch. 
How  much  did  each  cost? 

39.  A  and  B  do  a  piece  of  work  in  10  days.  A  does  66^  % 
as  much  work  as  B.  What  part  of  the  work  does  each  do  in 
10  days  ?  How  long  would  it  take  each  to  do  the  work  alone  ? 

40.  Henry's  money  equals  33  J  %  more  than  Harry's. 
What  is  the  ratio  of  Harry's  to  Henry's  ?  How  many  % 
is  Harry's  less  than  Henry's  ? 

41.  If  the  selling  price  of  goods  is  I  less  than  the  asking 
price,  the  J  is  equal  to  what  part  of  the  selling  price  ? 

42.  If  the  4-ft.  wood  which  a  dealer  sells  proves  to  be 
4  in.  short,  what  %  of  a  cd.  does  a  buyer  lose  ?  What  % 
does  the  dealer  make  ? 

43.  A  builder  who  charged  3%  for  superintending  the 
construction  of  a  bridge  received  $246  commission.  What 
was  the  actual  cost  of  the  bridge  ? 

44.  Of  a  mixture  of  milk  and  water  i  is  water.  How 
much  of  the  mixture  would  you  need  to  buy  to  obtain  a 
gal.  of  milk  ? 

If  4%  of  the  mixture  were  water,  how  much  must  be 
purchased  to  obtain  a  gal.  of  milk. 


ADVANCED    ARITHMETIC.  Ill 

45.  A  commission  merchant  received  3%  for  selling  and 
2%  for  buying.  After  selling  a  consignment  of  goods  and 
taking  out  both  commissions  he  invests  the  proceeds.  If 
the  entire  commission  is  $87,  what  amount  did  he  invest  ? 
What  is  the  ratio  of  the  3%  commission  to  the  amount 
after  deducting  it  ? 

1.  When  land  is  selling  at  an  advance  of  ^20  an  acre, 
what  is  the  gain  %  if  it  costs  $70  an  acre  ? 

2.  A  man  paid  an  agent  5%  for  selling  a  house  and 
received  $5824.  For  how  much  did  the  agent  sell  the 
house  ? 

3.  A  man  collects  debts  to  the  amount  of  $375.80. 
How  much  is  his  commission  at  4i%  ? 

4.  Mr.  Wilson  rents  160  acres  of  land  at  $2  an  acre. 
If  the  land  is  worth  $25  an  acre,  the  rent  equals  what  % 
of  the  value  of  the  land  ? 

5.  On  what  sum  do  taxes  at  J%  amount  to  $200  ? 

6.  If  an  investment  of  $624  yields  me  $43.68  per 
annum,  what  is  my  rate  of  profit  ? 

7.  A  man  bought  sugar  at  $9  per  cwt.  and  sold  it  at  a 
loss  of  4  % .     For  how  much  did  he  sell  it  ? 

8.  A  grocer  bought  8  doz.  eggs  at  10/  a  doz.  and  8  doz. 
at  5/  a  doz.  He  sold  them  all  at  15/  a  doz.  What  was 
his  %  of  gain  ? 

9.  If  oranges  are  bought  at  20/  and  sold  at  30/  a  doz., 
how  many  must  be  sold  for  a  man  to  realize  50%  on  an 
investment  of  $10  ? 

10.  If  by  selling  wine  at  $2.70  a  qt.  I  lose  10%,  at  what 
price  must  I  sell  it  to  gain  10%  ? 

11.  I  bought  $5000  worth  of  stock  at  95^.  At  what 
price  must  I  sell  it  to  gain  10%  ? 

12.  A  and  B  earned  $600.  A  is  to  receive  10%  less 
than  B.     Find  the  share  of  each, 


112  ADVANCED    ARITHMETIC. 

13.  If  I  lose  10%  by  selling  goods  at  40/  per  yd.,  at 
what  price  should  I  sell  to  gain  20%  ? 

14.  If  a  man  buys  a  house  for  $6840  and  receives  $1250 
for  rent  in  2  yr.  and  3  mo.,  what  rate  of  interest  does  the 
investment  yield  ? 

15.  If  I  sell  §  of  an  acre  of  land  for  an  amount  equal  to 
what  I  paid  for  1  acre,  what  %  do  I  make  ? 

16.  If  I  sell  I  of  a  quantity  for  an  amount  equal  to  what 
I  of  it  cost,  what  is  my  loss  %  ? 

17.  In  what  time  does  $600  at  7%  per  annum  produce 
$105.20  interest  ? 

18.  If  I  gain  40%  by  selling  wood  at  $6.40  a  cd.,  how 
much  did  it  cost  me  ? 

19.  Paid  $8000  for  stocks  at  16%  below  par  and  sold 
at  112.     What  %  was  gained  ? 

20.  A  man  performs  a  piece  of  work  in  8  days,  and  a 
boy  does  an  equal  amount  in  12  days.  The  amount  of 
work  the  boy  does  in  1  day  is  what  %  of  the  amount  the 
man  does  in  the  same  time  ? 

21.  A  man  sold  two  lots  for  $500  each,  gaining  20%  on 
one  and  losing  20%  on  the  other.  Did  he  gain  or  lose,  and 
how  much  ? 

22.  Bought  a  horse  and  carriage  for  $390 ;  the  cost  of 
the  carriage  was  equal  to  30%  of  the  cost  of  the  liorse. 
Find  the  cost  of  each. 

23.  For  what  sum  must  goods  worth  $6750  be  insured 
at  3^%  to  cover  both  property  and  premium  ? 

24.  What  was  received  for  a  sale  of  goods  marked  at 
$42.40,  at  6%  discount,  and  5%  oif  for  cash  ? 

25.  A  man's  taxes  are  $75.60,  and  this  equals  2i%  of 
the  value  of  his  property.     What  is  its  value  ? 

26.  The  number  of  peach  trees  in  an  orchard  is  20% 
more  than  the  number  of  apple  trees.  The  number  of  apple 
trees  is  how  many  %  less  than  the  number  of  peacli  trees  ? 


ADVANCED    ARITHMETIC.  113 

27.  An  agent  receives  $210  to  invest  in  corn  after  de- 
ducting his  commission  at  5%.  How  much  corn  does  he 
buy  at  50/  a  bu.  ? 

28.  An  agent  received  a  sum  of  money  to  invest  in  flour, 
after  taking  out  his  commission  at  8%.  If  he  invested 
^160  in  flour,  how  much  did  he  receive  ? 

29.  An  agent  sold  two  lots  for  $1000  each.  On  one  he 
gained  5%,  on  the  other  he  lost  5%.  Did  he  gain  or  lose, 
and  how  much  ? 

Interest.  —  1.  What  part  of  any  principal  equals  the 
interest  for  1  yr.  at  25%  ?  at  33^%  ?  at  7%  ?  at  10%  ? 
at  81%  ? 

What  is  the  ratio  in  each  case  of  the  principal  to  the 
interest  ? 

2.  How  many  hundredths  of  any  principal  equals  the 
interest  for  1  yr.  at  6%  ?  for  12  mo.  at  6%  ? 

What  part  of  any  principal  equals  the  interest  at  6% 
for  2  mo.  ?     Why  ? 

yi^  ^f  ^"y  pi*incipal  equals  the  interest  for  how  long  a 
time  at  6%  ? 

TOO  ^^  ^^^  equals  the  interest  of  $76  at  6%  for  how 
long  a  time  ? 

$.76  equals  the  interest  of  $76  for  2  mo.  at  6%. 

3.  Write  ten  sentences  similar  to  this  : 

T-Jo  of  $784  equals  the  interest  of  $784  at  6%  for  8  mo. 

4.  What  is  the  interest  of  $76  at  6%  for  9  mo.  ? 
What  equals  iihe  interest  at  6%  for  2  mo.  ?  for  9  mo.  ? 

^'      =?       oOf  r-r^of  $76  equals  the  interest  of  $76 

for  9  mo.  at  6%. 

5.  Make  many  sentences  like  this  :   y  of  yi^  of  $84 

11 '  %  84 
equals  the  interest  of  $84,  at  6%  for  11  mo. -^ —  =  ? 


114  ADVANCED    ARITHMETIC. 

What  is  the  interest  at  6%  of  — 

6.  $784  for  11  mo.  ? 

7.  $970  for  2  yr.  7  mo.  ? 

8.  $8604  for  1  yr.  3  mo.  ? 

9.  $126.96  for  3  yr.  6  mo.  ? 

10.  $2960  for  li  yr.  ? 

11.  $60.84  for  1  yr.  11  mo.  ? 

12.  $85.24  for  1  yr.  9  mo.  ? 

13.  $114  for  2  yr.  7  mo.  ? 

Practice  until  you  can  find  quickly  and  accurately  the  interest 
on  small  sums  for  short  periods. 

14.  In  each  of  the  above  what  is  the  ratio  of  interest  to 
principal  ?  of  principal  to  interest  ? 

Ans.  to  the  sixth  :  —  of  — — - ,  or  ,r-— ,  is  the  ratio  of  the 

interest   to   principal,     —r-   is   the   ratio  of   principal  to 
interest. 

1.  Assuming  30  da.  to  be  a  mo.,  how  many  days  in  .1 
of  a  mo.  ? 

2.  How  many  tenths  of  a  mo.  in  6  da.  ?  in  9  da.  ?  in 
15  da.  ?  in  7  da.  ?  in  16  da.  ?  in  28  da.  ? 

How  many  months  or  tenths  or  hundredths  months  in 
each  of  the  following  : 

3.  1  yr.  5  mo.  7  da.  ? 

1723 

1  yr.  5  mo.  7  da.  =  17.23  mo.,  or  mo. 

4.  5  mo.  12  da.  ? 

5.  6  mo.  15  da.  ? 

6.  1  yr.  6  mo.  6  da.  ? 

7.  3  yr.  7  mo.  9  da.  ? 

8.  2  yr.  5  mo.  17  da.  ? 

9.  1  yr.  9  mo.  19  da.  ? 


ADVANCED    ARITHMETIC.  115 

10.  What  is  the  interest  of  $225  for  2  yr.  7  mo.  11  da. 
at  6%? 

31  36'i2  25  ^^^*  ^^  *^^  interest  of  $225  at  6% 

— - — - — '■ — =  ?     for  1  mo.  ?     What,  then,  is  the  interest 
for  31.36  mo.  ? 

11.  Make  ten  sentences  similar  to  this :  U  of  J  of  y^^ 
of  $84  equals  the  interest  of  $84  for  7  mo.  18  da.  at  6%. 

=  the  interest  of  $84  at  6%   for  7  mo.  18  da. 

MVhat  is  the  interest  at  6%  of — 

12.  $245  for  6  mo.  18  da.  ? 

13.  $100  for  8  mo.  21  da.  ? 

14.  $960  for  3  yr.  7  mo.  9  da.  ? 

15.  $274  for  4  yr.  10  mo.  ? 

16.  $180  for  1  yr.  9  mo.  15  da.  ? 

17.  $967  for  1  yr.  4  mo.  17  da.  ? 

18.  $156.50  for  2  yr.  11  da.  ? 

19.  $980  for  3  yr.  9  mo.  13  da.  ? 

20.  $63.80  for  3  yr.  4  mo.  18  da.  ? 

21.  $460  for  2  yr.  6  mo.  19  da.  ? 

22.  $875  for  1  yr.  17  da.  ? 

23.  $897  for  1  yr.  1  mo.  4  da.  ? 

1.  A  note  draws  interest  at  6%  per  annum.  What 
is  the  ratio  of  one  year's  interest  to  the  principal  ?  of  the 
principal  to  one  year's  interest  ? 

2.  What  are  these  ratios  at  the  end  of  2  yr.  ?  3  yr.  ? 

2  yr.  6  mo.  ?    1  yr.  4  mo.  ?    1  yr.  10  mo.  ?   2  yr.  2  mo.  ? 

3  yr.  8  mo.  ?    2  yr.  3  mo.  ? 

Ans.  to  last.  The  interest  for  2  mo.  equals  yj^  of  prin- 
cipal ;  the  interest  for  2  yr.  3  mo.  equals  y  of  yj^,  or 
^7^  of  principal ;  ^j^  is  the  ratio  of  principal  to  interest. 

1  Find  first  what  equals  the  interest,  then  what  is  the  interest. 


116  ADVANCED   ARITHMETIC. 

3.  $54  is  the  interest  of  how  much  for  2  yr.  3  mo.  at 
6%? 

What  is  the  ratio  of  the  principal  to  the  interest  ?  of 

principal  to  the  amount  due  ?  of  interest  to  amount  ? 

100-2  .    ^-      ,.      ^^,         .     .     -. 
A71S.     — — —  IS  the  ratio  oi  the  principal  to  the  interest. 

^-—  is  the  ratio  of  the  principal  to  the  amount.     ^— -  is 

the  ratio  of  interest  to  amount. 

4.  $9761  is  the  amount  due  on  a  note  that  has  been 
drawing  interest  for  2  yr.  3  mo.  at  6%.  How  much  is  the 
principal  ?     How  much  is  the  interest  ? 

5.  What  is  the  ratio  of  the  interest  to  the  principal  for 
8  mo.  12  da.  at  6%  ?  of  principal  to  interest  ?  of  principal 
to  amount  ? 

6.  What  is  the  ratio  of  interest  to  tlie  principal  for 
7  mo.  9  da.  at  6%  ?  of  principal  to  interest?  of  principal 
to  amount  ? 

7.  What  is  the  ratio  of  interest  to  the  principal  for 
4  mo.  8  da.  at  6%  ?  of  principal  to  interest?  of  principal 
to  amount  ? 

8.  $612.81  is  the  amount  due  on  a  note  that  has  drawn 
interest  for  4  mo.  8  da.  at  6%.  What  is  the  face  of  the 
note  ?  the  interest  ? 

9.  $17.50  is  the  interest  of  $500  for  how  long  a  time 
at  6%  ? 

How  much  is  the  interest  for  2  mo.  ?  for  1  mo.  ?  What 
is  the  ratio  of  $17.50  to  the  interest  for  1  mo.  ?  What, 
then,  is  the  ratio  of  the  time  to  1  mo.  ?  The  time  is  how 
many  months  ? 

10.  $17  is  the  interest  of  $500  for  how  long  at 
6%  ? 

What  is  the  ratio  of  $17  to  the  interest  for  1  mo.  ?  6.8 
mo.  equals  6  mo.  and  how  many  days  ? 


ADVANCED    ARITHMETIC.  117 

11.  $31.20  is  the  interest  of  $400  for  how  long  at  6%  ? 
for  how  long  at  8%  ? 

12.  What  is  the  interest  of  $x  for  y  mo.  at  6%  ? 

13.  $(/-  is  the  interest  of  how  much  for  11  mo.  at  6<}^  ? 
for  b  mo.  ? 

14.  $c  is  the  amount  due  on  a  note  for  1  yr.  3  mo.  at 
6%  for  how  much  ? 

15.  $^  is  the  amount  due  for  m  mo.  at  6  % .     What  equals 
the  principal  ?  the  interest  ? 

1.  What  is  the  ratio  of  5%  to  6%  ?  of  8%  to  6%  ?  of 
9%  to  6%  ?  of  11%  to  6%  ?  of  71%  to  6%  ?  of  4i%  to 
6%  ?  of  81%  to  6%  ?  of  10%  to  6%  ?  of  4%  to  6%  ? 
of  7%  to  6%  ? 

Practice  until  these  ratios  can  be  thought  readily. 

2.  What  is  the  ratio  of  the  interest  of  any  principal  at 
8%  to  the  interest  for  the  same  time  at  6%  ? 

3.  What  is  the  ratio  of  the  interest  of  any  principal  at 
71%  to  the  interest  for  the  same  time  at  6%  ? 

4.  If  $x  is  the  interest  at  6%,  what  is  the  interest  for 
the  same  time  at  8%  ?  at  4%  ?  at  7^%  ?  at  9%  ? 

5.  What  is  the  interest  of  $242  at  9%  for  7  mo.  ? 

o  .  rr .  dt.o  40  What  equals  the  interest  of  $242  at  6% 

^T2~ — =  ^      for  7  mo. ?   What,  then,  equals  the  interest 

at9%? 

6.  Make  sentences  similar  to  this  : 

5   13.2   $7.84       the  interest  of  $784  for  1  yr.  1  mo.  6  da. 
4-2        ^~at7i%. 

Find  the  interest  of — 

7.  $360.75  for  3  yr.  3  mo.  at  7%. 

8.  $247.50  for  2  yr.  11  mo.  at  8%. 

9.  $843  for  11  mo.  27  da.  at  7^%. 
10.    $1055  for  10  mo.  27  da.  at  7^%. 


118  ADVANCED   ARITHMETIC. 

11.  $324  for  2  yr.  3  mo.  6  da.  at  4J%. 

12.  $675  for  11  mo.  9  da.  at  8%  ;  at  9%  ;  at  7J%  ;  10%. 

13.  What  is  the  amount  of  $840,  at  8%,  for  6  mo.  and 
24  da.  ? 

4  •  6.8  •  $8.40 


3-2 


+  $840  =  the  amount. 


Find  the  amount  of  — 

1.  $275  at  6%  from  May  2,  1882,  to  June  1,  1883. 

2.  $375  at  7%  from  Dec.  14,  1880,  to  May  5,  1881. 

3.  $242  at  7J%  from  Aug.  27,  1872,  to  July  12,  1880. 

4.  $85684  at  9%  from  Aug.  17, 1881,  to  April  4,  1882. 

5.  $783  at  5J%  from  Sept.  19,  1868,  to  June  1,  1870. 

6.  $3248  at  6%  from'  Jan.  4,  1882,  to  March  1,  1882. 

7.  $934  at  10%  from  Jan.  30,  1862,  to  Oct.  1,  1869. 

8.  $367.25  at  8J%  from  March  5,  1881,  to  Sept.  9, 
1882. 

9.  $79.42  at  6J%  from  Jan.  1,  1873,  to  Sept.  5,  1873. 

10.  $847.42  at  7%  from  May  5, 1881,  to  Aug.  11,  1881. 

11.  Solve  the  above  problems  again  more  quickly. 

12.  Compare  the  interest  of  $3  for  1  yr.  with  the  inter- 
est of  $3  for  6  mo. ;  for  3  mo. ;  for  1  mo. 

13.  How  much  money  will  earn  in  1  mo.  as  much  inter- 
est as  $500  earns  in  1  yr.  ? 

14.  How  much  money  must  a  man  have  on  interest  at 
6%  to  have  an  income  from  it  of  $10  a  da.,  counting  360 
da.  to  the  year  ? 

15.  How  much  must  I  invest  in  4%  bonds  to  receive 
from  them  an  income  of  $1000  a  year  ? 

16.  How  soon  will  interest  at  6  %  equal  the  principal  ? 
at5%?  at7i%? 

17.  I  had  $1000.  Loaned  i  of  it  at  4%  for  2  yr.  and 
the  remainder  at  5%  for  the  same  time.  How  much  inter- 
est did  I  receive  ? 


ADVANCED   ARITHMETIC.  119 

18.  Loaned  $500  at  6%  for  1  yr.  6  mo.  How  much 
more  interest  would  I  have  received  at  8%  ? 

19.  Mr.  B  had  $800.  He  loaned  f  of  it  for  a  year  at 
6%  and  the  balance  for  the  same  time  at  4%.  What  was 
the  amount  due  ? 

20.  What  is  the  ratio  of  the  interest  on  $300  for  4  yr. 
to  the  interest  on  $600  for  half  as  long  ? 

21.  What  ratio  does  the  interest  on  $300  for  1  yr.  bear 
to  the  interest  on  twice  the  principal  for  double  the  time, 
the  rate  being  the  same  ? 

22.  What  principal  will  earn  $50  in  1  yr.  4  mo.  at  6%  ? 

100_|50^^^26. 

O 

i|^  is  the  ratio  of  the  principal  to  the  interest. 

23.  What  principal  loaned  for  2  yr.  4  mo.  at  5%  will 
bear  $75  interest  ? 

^  f  of  $75  equals  interest  at  what  %  ? 

7T~r^ —  =  ^       W  ^s  ^^^  ^^^^^  ^^  principal  to  interest 

at  what  %  ? 

24.  The  interest  for  Ij  yr.  at  4%  is  $100.  Find  the 
principal. 

25.  The  interest  for  4  yr.  at  8%  is  $400.  Find  the 
principal. 

26.  What  principal  will  bear  $180  interest  in  3J  yr.  at 
7%  ?     At  6%  the  principal  would  yield  what  interest  ? 

1.    At  6%  per  annum,  what  part  of  the  principal  equals 
the  interest  for  2  yr.  7  mo.  12  da.  ?  at  8%  ?  at  ^%  ? 
.12  What  part  of  the  principal  equals  the  interest 

.035         for  2  yr.  ? 

'^^^  What  part  of  the  principal  equals  the  interest 

.157         for  7  mo.?  Ans.    |  of  ^^  =  i^Jtt  =  tMtt- 

What  part  of  the  principal  equals  the  interest  for  12  da.  ? 
Ans.    ^^  of  J  of  jIjj  =  jjj^^  =  i^jj' 


120  ADVANCED    ARITHMETIC. 

At  6  %  per  annum,  what  part  of  the  principal  equals  the 
interest  for  — 

2.  2  yr.  4  mo.  ?  9.  5  mo.  21  da.  ? 

3.  lyr.  7mo.?  10.  11  mo.  14  da.  ? 

4.  5  yr.  8  mo.  ?  11.  5  mo.  29  da.  ? 

5.  2yr.  9mo.?  12.  1  yr.  6  mo.  6  da.? 

6.  3  yr.  11  mo.  ?  13.  2  yr.  7  mo.  9  da.? 

7.  7  mo.  7  da.  ?  14.  1  yr.  11  mo.  19  da.? 

8.  9  mo.  7  da.?  15.  4  yr.  4  mo.  4  da.  ? 

16.  At  6%  per  annum,  what  part  of  $840  equals  the 
interest  for  2  yr.  7  mo.  ? 

17.  How  much  is  the  interest  on  $840  for  1  yr.  7  mo.  at 

Exact  Interest. — To  find  the  exact  interest  for  a  part  of 
a  year,  reckon  365  days  to  the  year. 

1.  The  interest  of  any  principal  for  52  da.  equals  how 
many  365th s  of  the  interest  for  1  yr.  ? 

2.  What  is  the  exact  interest  of  $750  for  70  da.  at  7%  ? 

70  *  7  *  *lj>7  ^0 

J.  J         =  the  exact  interest  of  $750  at  7%  for  70  da. 
obo  ' 

Find  the  exact  interest  of  the  following : 

3.  $245  for  40  da.  at  8%.    • 

4.  $95  for  75  da.  at  6%. 

5.  $840  for  93  da.  at  9%. 

6.  $743  for  45  da.  at  7^%. 

7.  $366  for  63  da.  at  6%. 

8.  $244  for  33  da.  at  8%. 

9.  $727  for  85  da.  at  10%. 

10.  What  is  the  exact  interest  on  $1000  from  March  1, 
1885,  to  April  19,  1886,  at  4%  ? 

11.  What  is  the  exact  interest  of  a  note  for  $95,  dated 
Jan.  10,  1891,  due  July  1,  1891  ? 


ADVANCED    ARITHMETIC.  121 

Compound  Interest.  —  1.  What  is  the  compound  interest 
of  $240  for  2  yr.  6  mo.  at  6%  ? 

What  is  the  amount  of  $240  for  1  yr.  at  6%  ?     . 

What  is  the  amount  of  $254.40  for  1  yr.  at  6%  ? 

What  is  the  amount  of  $269.06  for  6  mo.  at  6%  ? 

What  is  the  difference  between  $240  and  $269.66  ? 

The  difference  between  the  last  amount  and  the  first 
principal  is  the  compound  interest  of  $240  for  2  yr.  6  mo. 
at  6%. 

State  what  you  did  in  finding  the  compound  interest  of 
$240  for  2  yr.  6  mo.  at  6%. 

What  is  compound  interest  ? 

How  is  compound  interest  found  ? 

2.  What  is  the  amount  of  $350  for  5  yr.  at  4<^  com- 
pounded annually  ? 

3.  What  is  the  compound  interest  of  $570  for  2  yr. 
8  mo.  at  10%? 

4.  What  is  the  amount  of  $600  at  8%  for  1^  yr.  com- 
pounded semi-annually  ?  How  many  interest  periods  are 
there  ?  If  8  %  is  the  rate  for  1  y r.,  what  is  the  rate  for 
6  mo.  ? 

5.  What  is  the  compound  interest  of  $92  at  T'J^  for 
2  yr.  4  mo.  6  da.  compounded  semi-annually  ?  How  many 
whole  intervals  of  time  in  the  above  problem  ?  How 
many  months  added  in  the  fractional  interval? 

6.  What  part  of  $1  equals  the  compound  interest  of 
$1  for  3  yr.  at  8%  ? 

What  partj,  then,  of  $327  equals  the  compound  interest 
of  $327  for  3  yr.  ? 

7.  What  is  the  amount  at  compound  interest  of  $1  for 
4  yr.  6  mo.  at  7%  ?  What  is  the  amount  of  $675  for  the 
same  time  and  rate  ? 

8.  What  is  the  amount  at  compound  interest  of  $675 
for  4  yr.  6  mo.  at  4%  ? 


122  ADVANCED    ARITHMETIC. 

Find  the  amount  of  each  of  the  following  at  compound 
interest : 

9.    $745  for  3  yr.  at  6%. 

10.  $1020  for  4  yr.  at  6%. 

11.  $28.45  for  3iyr.  at  7%. 

12.  $100  for  3  yr.  8  mo.  at  8%. 

13.  $520  for  2  yr.  9  mo.  at  8%. 

14.  $740  for  4  yr.  6  mo.  at  7%. 

15.  If  the  compound  interest,  rate,  and  time  are  given, 
how  find  the  principal  ? 

Annual  Interest.  —  If  a  note  contains  the  words  "with 
annual  interest,"  or  "interest  payable  annually,"  the  interest 
is  due  at  the  close  of  each  year,  and,  if  not  paid  when  due, 
this  "  annual  interest "  draws  simple  interest  until  paid. 

1.    Find  the   amount  of   $500   for  4  yr.  6  mo.  with 
interest  payable  annually  at  8%,  interest  unpaid. 

What  interest  is  due  on  the  principal  annually  ? 

What  is  the  total  annual  interest  ? 

For  what  time  does  the  first  annual  interest  draw 
interest  ?   the  second  ?   the  third  ?   the  fourth  ?  ^ 

The  interest  of  $40  for  3  yr.  6  mo.,  2  yr.  6  mo.,  1  yr. 
6  mo.  and  for  6  mo.  is  equal  to  the  interest  of  $40  for 
what  time  ? 

The  interest  of  $40  for  8  yr.  at  8%  is  how  much? 

What  is  the  amount  of  the  annual  interest,  interest  of 
the  annual  interest,  and  the  principal  ? 

.08  of  $500  =  $40.     Interest  due  annually. 

41-  •  $40  =  $180.     Total  annual  interest. 

8  •  .08  of  $40  =  $25.60.    Simple  interest  of  $40  for  8  yr. 
$500 
180 
25.60 


$705.60     Amount  due. 


ADVANCED    ARITHMETIC.  123 

2.  At  the  end  of  3  yr.,  what  is  due  on  a  debt  of  $400 
with  interest  annually  at  7  %  ? 

3.  In  what  respect  does  annual   interest   differ  from 
compound  interest? 

Partial  Payments.  —  1.  The  following  is  a  promissory  note  : 
$200.  Chicago,  June  4,  1889. 

Two  months  after  date,  I  promise  to  pay  James  Bergh 
two  hundred  dollars  for  value  received,  interest  6%. 

Henry  Pallas. 

(a)    Who  is  the  maker  of  the  above  note  ? 
(h)    To  whom  is  the  note  made  payable,  or  who  is  the 
payee  ? 

(c)  How  much  does  the  maker,  or  drawer  of  the  note 
promise,  or  what  is  the  face  of  the  note  ? 

(d)  Who  is  the  holder  or  owner  of  the  note  ? 

(e)  What  is  its  date  ? 

(/)    When  is  the  note  to  be  paid  ? 

(g)    Tell  as  many  things  as  you  can  about  the  note. 

(Ji)    What  is  a  promissory  note  ? 

(i)    What  is  the  face  of  the  note  ? 

(a) 
$600.  Englewood,  III.,  June  28,  1888. 

On  demand  I  promise  to  pay  to  William  Jones  six  hun- 
dred dollars  with  interest  at  7%.     Value  received. 

Edward  Dale. 

$600.  Englewood,  III.,  June  28,  1888. 

One  year  after  date,  I  promise  to  pay  William  Jones, 
or  order,  six  hundred  dollars  with  interest  at  7%.  Value 
received.  Edward  Dale. 


124  ADVANCED    ARITHMETIC. 

$600.  Englewood,  III.,  June  28,  1888. 

One  year  after  date,  I  promise  to  pay  to  William  Jones 
or  bearer  six  hundred  dollars  with  interest  at  7%.  Value 
received.  Edward  Dale. 

(a)  In  what  respect  are  the  notes  a,  h,  and  c  alike  ? 

(b)  How  does  note  a  differ  from  note  b  ? 

(c)  How  soon  after  date  is  note  a  payable  ?  note  b  ? 

(d)  To  whom  is  note  a  payable?  note  b?  Is  note  b 
payable  to  any  one  else  ? 

(e)  How  may  note  b  be  transferred  to  some  one  else  ? 
(/)  If  William  Jones  should  write  on  the  back  of  note  b, 

"  Pay  to  John  Condit, 

William  Jones," 
to  whom  would  the  note  be  payable  ? 

(g)  How  could  William  Jones  make  the  note  payable  to 
any  one  ? 

(/i)    If  William  Jones  should  write  on  the  back  of  note  b, 
"  Pay  to  John  Condit  or  order," 
to  whom  could  John  Condit  transfer  the  note  ? 

(i)    How  does  note  c  differ  from  note  b  ? 

(j)    Explain  the  essential  differences  in  the  three  notes. 

(k)  Why  should  "  value  received "  be  written  in  each 
note  ? 

(I )  Draw  a  demand  note.  Draw  a  time  note  payable  to 
order ;  another  to  bearer. 

When  a  part  of  a  debt  is  paid,  the  amount  and  date  of 
the  payment  are  written  on  the  back  of  the  note.  This  is 
called  an  indorsement.     What  is  an  indorsement  ? 

The  following  is 'the  United  States  method  of  finding 
amount  due  upon  a  note  on  which  partial  payments  have 
been  made.     Find  the  amount  of  the  given  principal  to  a 


ADVANCED    ARITHMETIC.  125 

time  when  a  payment  or  the  sum  of  the  payments  is  equal 
to  or  greater  than  the  interest  then  due.  Deduct  the  pay- 
ment or  the  sum  of  the  payments  from  the  amount.  With 
the  remainder  as  a  new  principal  proceed  as  before. 

1.    $200.  Auburn  Park,  April  14,  1872. 

On  demand,  for  value  received,  I  promise  to  pay  to  Ford 
McKnight,  or  order,  two  hundred  dollars  with  interest  at 

1^'  James  Hughes. 

Partial  payments  were  indorsed  on  this  note  as  follows  : 
Oct.  14,  1872,  $46 ;  April  14,  1874,  $10 ;  June  14,  1875, 


What  was  the  amount  due  Jan.  26,  1877  ? 

First  principal. 

$200.00 

Interest  to  Oct.  14,  1872,  6  mo., 

6.00 

Amount, 

$206.00 

First  payment, 

46.00 

Second  principal, 

$160.00 

The  payment  ($10)  made  April  14, 1874,  is  less  than  the 
interest  ($14.40)  due  at  that  date  ;  hence  the  interest  is 
reckoned  to  June  14,  1875. 

Interest  to  June  14,  1875,  2  yr.  8  mo.,  $25.60 

Amount,  $185760 

Sum  of  second  and  third  payments,  $60.00 

Third  principal,  $125.60 

Interest  to  Jan.  26,  1877,  1  yr.  7  mo.  12  da.,  12.18 

Amount  due  Jan.  26,  1877,  $137.78 

2.    A  note  of  $500  is  dated  March  10,  1873. 

Indorsements  : 
Nov.  10,  1873,  $75;  July  10,  1874,  $100, 
What  was  due  May  10,  1876,  at  6%  interest  ? 


126  ADVANCED   ARITHMETIC. 

3.  A  note  of  $100  dated  Jan.  9,  1880. 

Indorsements  : 
Oct.  15,  1881,  $250 ;  Dec.  27,  1883,  $25 ;  June  18,  1884, 
$350. 

What  was  due  July  13,  1885,  at  6%  interest? 

4.  A  note  of  $700  dated  March  12, 1883,  is  indorsed  as 
follows  : 

June  10,  1883,  $45 ;  July  27,  1885,  $24  ;  Oct.  15,  1886, 
$200  ;  Nov.  13,  1887,  $190. 

What  was  due  Oct.  9,  1888,  interest  8%  ? 

Notes  and  accounts  running  for  one  year  or  less  are  often 
settled  by  what  is  known  as  the  Mercantile  Method. 

The  method  is  as  follows : 

Find  the  amount  of  the  note  or  other  obligation  at  the 
time  of  the  settlement,  and  of  each  payment  from  the  time 
it  was  made  until  the  date  of  settlement. 

From  the  amount  of  the  note  or  other  obligation,  subtract 
the  sum  of  the  amounts  of  the  payments. 

5.  Solve  the  last  two  problems  by  the  Mercantile 
Method.  Which  method  is  more  favorable  to  the  creditor  ? 
Which  seems  more  just  ?     Why  ? 

6.  A  note  of  $850  dated  Feb.  6,  1870,  and  drawing 
interest  at  7^%,  was  indorsed  as  follows  : 

May  15, 1870,  $270  ;  Sept.  12,  1870,  $80 ;  Nov.  20,  1870, 
$120. 

What  was  due  Jan.  1,  1871  ? 

7.  On  a  note  for  $740  at  10^^  interest  dated  Feb.  1, 
1887,  and  maturing  10  months  after  date,  the  following 
indorsements  were  made : 

April  15,  1887,  $60  ;  May  3,  1887,  $85 ;  Sept.  12,  1887, 
$300. 

What  was  the  balance  due  at  the  time  of  payment  ? 


ADVANCED    ARITHMETIC.  127 

Present  Worth.  —  1.  If  x  is  the  amount  of  money  a  farmer 
had  after  a  gain  of  \^  what  equals  the  sum  on  which  he 
gained  the  \  ? 

2.  What  is  a  man  worth  if  $535  is  .08  more  than  what 
he  is  now  worth  ? 

3.  If  the  future  value  of  a  lot  is  $990,  which  is  10% 
more  than  its  present  worth,  what  part  of  $990  equals  the 
present  worth  ? 

4.  If  %x  were  the  amount  of  money  a  man  agreed  to 
pay  you  at  the  end  of  a  year,  and  money  were  worth  20% 
to  you,  what  would  you  be  willing  to  take  for  the  debt 
to-day  ? 

If  you  add  J  or  20%  to  what  you  are  willing  to  take, 
does  the  sum  equal  %x  ? 

What  you  are  willing  to  take  equals  how  many  sixths 

of  $iC? 

5.  How  much  cash  is  equivalent  to  a  claim  of  $224 
payable  in  2  yr.,  provided  the  use  of  the  money  is  worth 
6  %  per  annum  ? 

$200  is  the  present  worth  of  the  above  debt ;  what  is 
the  present  worth  of  any  debt  payable  at  a  future 
time? 

$24  is  the  true  discount ;  what  is  the  true  discount  of 
any  debt  ? 

6.  What  ought  I  to  receive  for  a  note  of  $275,  which 
bears  no  interest,  and  which  will  become  due  in  2  yr.  4  mo., 
the  use  of  the  money  being  worth  9%  per  annum  ? 

100   $275     ^    ^ 
121       ~  • 

7.  What  is  the  difference  between  the  true  discount 
and  the  interest  of  $317  for  3  yr.  at  7%  ? 

8.  What  is  the  face  of  a  note  due  in  6  mo.,  with 
interest  at  8%,  that  will  cancel  a  debt  of  $232  due  in 
6  mo.  without  interest  ? 


128  ADVANCED    ARITHMETIC. 

9.  What  is  the  difference  in  value  between  a  cash  pay- 
ment of  $225  and  a  note  of  $230  due  in  8  mo.  without 
interest,  the  use  of  money  being  worth  7J%  ?  if  the  use  of 
money  is  worth  5%  ? 

Borrowing  Money.  —  The  systems  of  loaning  money  at  a 
bank  and  by  private  individuals  are  not  alike. 

If  A  borrows  $300  of  B  for  1  year,  at  10%,  when  does 
A  pay  the  interest  ?  What  amount  of  money  does  A  pay 
B  at  the  end  of  the  year  ? 

If  A  wishes  to  borrow  about  $300  for  2  mo.  at  a  bank, 
he  makes  a  note  for  $300,  without  interest,  and  presents  it, 
properly  secured.  The  banker  computes  the  interest  on  the 
face  of  the  note  for  63  da.,  and  deducts  this  amount  from 
the  face,  paying  A  the  difference. 

If  A  holds  an  interest-bearing  note  against  B  for  $300 
due  in  1  yr.,  at  10%,  and  wishes  to  obtain  money  by  sell- 
ing that  note  at  a  bank,  he  presents  it,  properly  secured. 
The  banker  deducts  interest  for  3  da.  more  than  the  specified 
time  from  the  amomit  of  the  note  ($330)  instead  of  from 
its  face. 

The  amount  a  banker  deducts  is  called  hank  discount.  It 
equals  the  interest,  as  just  described,  of  the  amount  that  he 
can  collect  on  the  note  for  three  days  more  than  the  time 
specified. 

The  sum  paid  after  deducting  the  bank  discount  is  called 
proceed^. 

The  three  days  more  than  the  specified  time  are  called 
days  of  grace. 

A  note  matures,  or  is  legally  due,  on  the  last  day  of 
grace.  If  the  last  day  falls  on  Sunday  or  a  legal  holiday, 
it  is  due  the  day  before. 

In  what  ways  do  the  systems  of  borrowing  money  at  a 
bank  and  of  a  private  individual  differ  ? 


ADVANCED   ARITHMETIC.  129 

Suppose  A  gives  a  bank  a  note  for  $300  and  receives  '|294  as 
the  proceeds ;  how  much  does  the  bank  reserve  as  interest  ? 

The  $300  is  the  sum  of  the  proceeds  and  what  ? 

The  $6  is  the  interest  on  the  proceeds  and  on  what  else  ? 

What  does  a  bank  gain  by  deducting,  at  the  time  the  loan  is 
made,  interest  on  the  face  or  the  amount  of  a  noteV 

In  addition  to  obtaining  interest  on  interest,  what  other  advan- 
tage does  the  bank  gain  ? 

Wlien  A  borrows  of  B,  who  has  the  use  of  the  interest  until  the 
debt  is  paid  ?     When  A  borrows  of  a  bank  ? 

What  does  a  bank  receive  at  the  date  of  maturity  of  a  non-in- 
terest-bearing note  ?     On  an  interest-bearing  note  ? 

On  M'hat  is  the  bank  discount  reckoned  in  dealing  with  a  non- 
interest-bearing  note?  On  what,  then,  should  it  be  reckoned  in 
dealing  with  an  interest-bearing  note? 

Does  a  man  who  borrows  money  at  a  bank  pay  interest  for  the 
three  days  of  grace  ? 

What  does  the  debtor  gain,  if  anything,  by  being  allowed  three 
days  of  grace  ? 

Explain  the  system  of  borrowing  money  at  a  bank  on  an 
interest-bearing  and  on  a  non-interest-bearing  note. 

What  does  a  bank  gain  by  deducting  interest  on  the  face 
or  amount  of  the  note  at  the  time  the  loan  is  made  ? 

1.  What  is  the  bank  discount  of  a  note  of  $150 
payable  in  60  da.,  discounted  at  8%  ?  What  are  the 
proceeds  ? 

What  part  of  the  face  of  the  note  equals  the  discount 

for  63  da.  ? '     ' —  =  bank  discount. 

Find  the  bank  discount  and  the  proceeds  of  a  note  of  — 

2.  $100  payable  in  60  da.,  discounted  at  6%. 

3.  $75  payable  in  90  da.,  discounted  at  10%. 

4.  $25.50  payable  in  30  da.,  discounted  at  8%. 

5.  $727  payable  in  100  da.,  discounted  at  10%. 


130  ADVANCED    ARITHMETIC. 

6.  $50  dated  June  1,  1875,  payable  Sept.  1,  1875,  dis- 
counted at  10%. 

What  is  the  actual  time  in  days  between  June  1  and 
Sept.  1  ? 

7.  $170  dated  March  7,  1887,  payable  July  9,  1887, 
discounted  at  9%. 

8.  $11.75  dated  Jan.  1,  1889,  payable  March  1,  1889, 
discounted  at  9%. 

9.  What  is  the  difference  between  the  true  discount 
and  the  bank  discount  of  $874  due  in  100  da.,  discounted 
at  6%  ? 

10.  What  is  the  difference  between  the  true  discount  and 
the  bank  discount  of  $95,  due  in  57  da.,  discounted  at  8%  ? 

11.  What  is  the  date  of  maturity  of  a  note,  given  May 
10,  1884,  and  payable  in  90  da.  ?    * 

Note.  —  To  find  the  date  of  maturity  when  the  time  is  ex- 
pressed in  days,  count  forward  for  three  days  more  than  the  given 
number  of  days.  When  the  time  is  expressed  in  a  certain  number 
of  months,  the  note  is  nominally  due  on  the  corresponding  day  of 
the  month,  or,  when  there  is  no  corresponding  day,  on  the  last  day 
of  the  month. 

A  2-mo.  note  dated  Dec.  31, 1888,  is  due  Feb.  28.  When 
is  a  3-mo.  note  dated  Dec.  31,  1888,  due  ? 

When  is  a  2-mo.  note  dated  July  31,  1857,  due  ? 

12.  A  note,  dated  Jan.  7,  1885,  was  payable  in  60  da. ; 
what  was  the  date  of  maturity  ?  If  the  note  had  been 
payable  in  2  mo.,  what  would  have  been  the  date  of  ma- 
turity ? 

13.  If  $1  were  the  proceeds  of  a  note  out  of  which  .1 
had  been  taken,  $1  would  equal  what  part  of  the  face  of 
the  note  ? 

14.  If  $250  were  the  proceeds  of  a  note  out  of  which 
.06  had  been  taken,  $250  would  equal  how  many  hun- 
dredths of  the  face  of  the  note  ? 


ADVANCED    ARITHMETIC.  131 

15.  If  m  dollars  were  the  proceeds  of  a  note  out  of  which 
.03  had  been  taken,  m  dollars  would  equal  what  part  of  the 
face  of  the  note  ? 

16.  If  $450  were  the  proceeds  of  a  note  which  had 
been  discounted  6%,  what  would  be  the  face  of  the  note  ? 

17.  For  what  sum  must  a  note,  payable  in  57  da.,  be 
drawn  to  produce  $95  when  discounted  at  6%  ? 

18.  What  is  the  face  of  a  90-day  note  which,  discounted 
at  7%,  will  give  as  proceeds  $275  ? 

1.  What  is  the  cost  of  a  sight  draft  on  Chicago  for 
$485  at  1%  premium? 

2.  What  is  the  cost  of  a  draft  for  $2840,  discount  |%  ? 
Find  the  cost  of  drafts  for  the  following  amounts  : 

3.  $564,  premium  i  %.  6.    $475,  discount  |%. 

4.  $2700,  premium  J%.  7.    $185,  discount  1^%. 

5.  $6000,  premium  J^.  8.    $837,  discount  |%. 
9.    If  the  cost  of  a  sight  draft  at  |  %  premium  is  $873, 

what  is  the  face  of  the  draft  ? 

Find  the  face  of  a  draft  which  cost  — 

10.  $620,  premium  li-%.  12.    $2400,  discount  i%. 

11.  $396,  premium  |%.  13.    $847,  discount  |%. 

14.  Find  the  cost  of  a  draft  of  $560  payable  in  4  mo., 
interest  at  7%  allowed  until  the  draft  is  paid,  premium  |%. 

What  would  the  draft  cost  if  no  deduction  was  made  for 
the  interest  ? 

What  is  the  interest  of  $560  at  7%  for  4  mo.  ? 

$564.20  less  the  interest  equals  what  ? 

Find  the  cost  of  drafts  for  the  following  amounts : 

15.  $285,  premium  1^%,  time  60  da.,  interest  4%. 

16.  $1000,  premium  |%,  time  2  mo.,  interest  6%. 

17.  $2840,  premium  J%,  time  90  da.,  interest  5%. 

18.  $700,  discount  |%,  time  3  mo.,  interest  4J%. 

19.  What  is  the  face  of  a  draft  bought  in  New  York^ 


132  ADVANCED    ARITHMETIC. 

which  will  pay  a  debt  in  Chicago  of  $2500,  exchange  on 
New  York  being  \(']q  premium  in  Chicago  ? 

20.  A  commission  merchant  in  Chicago  pays  by  draft  a 
merchant  in  Kansas  City  a  debt  of  $480,  drafts  on  Chicago 
being  at  a  premium  of  j  %  in  Kansas  City.  What  is  the 
face  of  the  draft  which  pays  the  debt  ? 

21.  An  agent  in  Davenport  sold  a  consignment  of  goods 
for  %1'^^^,  commission  at  the  rate  of  2%.  He  remitted 
the  proceeds  by  draft  on  Chicago  at  a  premium  of  |%. 
What  was  the  amount  remitted  ? 


MENSURATION. 

"  Geometry  reduces  its  laws  and  attributes  of  magnitude  to  per- 
fect clearness  by  according  to  the  senses  a  representation  of  those 
lines,  surfaces,  and  solids  which  it  conceives  with  the  utmost  com- 
pleteness and  precision ;  and  thus,  issuing  forth  from  behind  the 
veil  of  mental  invisibility  into  the  visible  and  palpable,  its  doc- 
trines may  almost  be  seen  and  handled,  and  yet  without  losing 
aught  of  their  purity  and  necessity.  Thus,  geometry,  if  I  may 
so  express  myself,  becomes  a  thinking  with  the  eye.  .  .  .  This  rela- 
tion of  its  laws  to  determinate  figures ;  this  apprehension  of  the 
highest  and  most  surprising  doctrines  through  the  visibility  of 
body  is  precisely  what  at  once  attracts  and  animates  the  young  — 
what  gradually  elevates  and  prepares  for  high  abstraction  their 
powers  as  yet  incapable  of  such  exercise." — Thiersch. 

"  Unless  the  definitions  are  intuitions  of  the  figures  and  rela- 
tions defined,  they  are  also  barren.  .  .  .  The  mathematician  does 
not  begin  by  assuming  the  properties  of  figures,  and  after  defining 
them  proceed  to  ascertain  whether  such  figures  exist ;  he  begins 
by  ascertaining  that  such  figures  and  such  relations  do  exist,  and 
then  defines  them  as  he  finds  them.  In  other  words,  definitions 
are  the  expressions  of  the  figures,  not  their  foundations."  —  G.  11. 
Lewes. 

"  So  far  as  the  student  has  the  power  let  him  seize  tlie  most 
general  point  of  view  within  his  reach,  .  .  .  but  let  him  shun  the 
common  fault  of  taking  indistinctness  as  evidence  of  generaliza- 
tion, for  this  is  to  suppose  ourselves  on  a  mountain  because  we  are 
in  a  mist." — Dr.  Whewell. 

"  Gauss,  whose  authority  on  such  a  subject  weighs  against  a 
whole  academy,  declared  geometry  to  be  the  '  science  of  the  eye  ' ; 
and  Professor  Sylvester,  also  a  very  considerable  authority,  de- 
clares that  most  if  not  all  the  great  ideas  of  modern  mathematics 

133 


134 


ADVANCED   ARITHMETIC. 


ADVANCED   ARITHMETIC.  135 

had  their  origin  in  observation.  .  .  .  Even  in  the  higher  develop- 
ments of  the  Calculus,  where  sensible  experiences  seem  most  widely 
departed  from,  it  is  easy  to  trace  a  sensible  origin  for  the  extra 
sensible  intuitions.  .  .  .  Most  of  the  difficulties  in  this  science  are 
difficulties  rather  of  intuition  than  of  reasoning." — Geo.  H.  Lewes. 

"  The  higher  processes  of  mind  in  mathematics  lie  at  the  very 
foundation  of  the  subject."  —  Joseph  Sylvester. 

Solids,  Surfaces,  and  Lines.  —  1.  In  how  many  directions 
can  you  extend  your  hand  ? 

In  how  many  directions  does  a  sphere  extend  from  any 
point  within  it  ?  an  ovoid  ?  a  prism  ?  a  cylinder  ?  a  lump 
of  coal  ?  a  cone  ?  a  cube  ? 

In  how  many  directions  does  the  air  of  this  room  extend 
from  any  point  within  ? 

2.  How  far  can  we  think  extension  in  any  one  direc- 
tion ?  How  far  can  we  think  extension  in  every  direction  ? 
Of  what  can  you  think  that  is  limited  in  its  extent  in  every 
direction  ? 

Name  a  solid,  that  is,  something  whose  extension  is 
limited  in  every  direction. 

Think  whether  a  cylinder,  an  orange,  the  air  in  the 
room,  the  water  in  a  tumbler  can  be  called  solids. 

Give  other  examples  of  portions  of  space  limited  in  every 
direction. 

3.  What  limits  the  extension  of  an  apple  ?  of  a  sphere  ? 
of  a  pencil  ?  of  a  brick  ?     What  is  a  surface  ? 

Here  is  a  glass  partly  full  of  water.  What  is  in  contact 
with  the  upper  surface  of  the  water  ?  Has  the  air  in  the 
glass  a  surface  as  well  as  the  water  ?  Has  the  air  one 
surface  and  the  water  another  ?  What  limits  or  bounds 
the  extension  of  the  water  ?  Then  what  is  the  boundary 
of  the  water  ?  What  limits  the  extension  of  the  air  in  this 
room  ?     Then  what  is  the  boundary  of  this  body  of  air  ? 


136  ADVANCED    ARITHMETIC. 

What  limits  the  extension  of  a  sphere  ?  of  a  cube  ?  of  any- 
solid  ? 

Of  how  many  smaller  surfaces  is  the  entire  surface  of  a 
cube  composed  ?  Show  me  the  boundary  of  one  of  these 
smaller  surfaces. 

The  surface  of  a  hemisphere  is  composed  of  how  many 
smaller  surfaces  ?  What  limits  the  extension  of  tlie  plane 
surface  ?  of  the  curved  surface  ? 

Move  your  hand  along  the  edge  or  boundary  of  the  top 
of  your  desk;  along  the  boundary  of  the  upper  surface  of 
the  table. 

Think  of  a  square  piece  of  paper.  Think  of  the  edges 
or  limits  of  its  extension.  Think  of  the  edge  or  boundary 
of  the  surfaces  of  different  objects. 

What  is  an  edge  ? 

4.  Place  a  finger  on  a  corner  of  the  cube.  In  how 
many  directions  from  one  of  its  corners  do  the  edges  of  a 
cube  extend  ? 

Think  of  the  extension  of  each  of  these  edges. 

Find  the  upper  right  corner  in  the  front  of  the  room. 
In  how  many  directions  from  this  corner  do  the  edges  of 
the  room  extend  ? 

In  how  many  directions  from  the  corner  of  the  black- 
board do  its  edges  extend  ?  Think  of  the  extension  in  07ie 
direction  of  an  edge  of  a  cube.  Of  what  kind  of  a  line  did 
you  think  ? 

TJiink  of  extension  in  one  direction. 

What  is  a  straight  line  ? 

What  kind  of  a  line  do  you  think  when  your  mind  moves 
along  an  edge  of  a  cube  ?  of  a  pane  of  glass  ? 

Is  there  any  difference  between  an  edge  and  a  line  ? 

Name  things  that  represent  or  suggest  straight  lines,  or 
extension  in  one  direction. 

5.  What  is  found  at  each  end  of  one  of  the  edges  of  a 


ADVANCED    ARITHMETIC. 


137 


cube  ?     How  many  lines  meet  or  end  at  this  point  ?    Does 
a  point  have  extension  ? 

What  is  a  point  ? 

A  point  is  position. 

What  is  used  to  represent  a  point  ? 

Note. —  Pupils  should  be  able  to  state  questions  before  trying  to 
answer  them. 

6.    Observe  the  curved  surface  of  a  cylinder.     Can  two 
points    be   selected  in  the  curved   surface  of   a  cylinder 


that  cannot  be  connected  by  a  straight  line  lying  wholly 
within  this  surface  ? 

If  the  mind  thinks  one  unchanged  direction  in  passing 
from  a  to  n,  does  the  line  an  lie  within  the  surface  of  the 
cylinder  ? 

Can  two  points  be  so  placed  in  the  curved  surface  of  the 
cylinder  that  they  may  be  connected  by  a  straight  line 
lying  wholly  in  the  surface  ?  Can  any  two  points  what- 
soever in  the  surface  of  the  blackboard  be  connected  by  a 
straight  line  lying  wholly  in  its  surface  ?  Try  to  find  two 
that  cannot  be  so  connected. 


138  ADVANCED    ARITHMETIC. 

Are  there  any  two  points  in  the  top  of  this  desk  that 
cannot  be  so  connected  ? 

The  blackboard  is  a  plane,  so  is  the  top  of  this  desk. 
How  does  a  plane  differ  from  a  curved  surface  ? 

What  is  a  plane  ? 

A  plane  is  a  surface  any  two  points  oftvhich  may  he  joined 
by  a  straight  line  lying  wholly  within  the  surface. 

7.    What   is  extension  in  every  direction  ?     What  is 
extension  in  one  direction  ?     What  is  a  solid  ? 

What  is  the  boundary  of  a  solid  ?  The  beginning  and 
the  limit  of  extension  in  one  direction  are  called  what  ? 

Make  a  drawing  to  represent  extension  in  one  direction. 
What  is  a  straight  line  ?  Why  do  not  the  edges  of  a  cyl- 
inder suggest  straight  lines  ?  Do  the  edges  of  a  cube  ? 
Why  ?     Represent  a  point. 

What  is  a  point  ?  Why  is  the  intersection  of  two  lines 
a  point  ?     Does  a  point  have  extension  ? 

What  is  a  plane  ? 

Relations  of  Magnitude.^  —  Find  equal  solids.  Eind  un- 
equal solids.  Find  equal  surfaces.  Find  unequal  surfaces. 
Find  equal  lines ;  unequal  lines. 

What  relations  of  magnitude  may  solids  have  ?  surfaces  ? 
lines  ? 

Relations  of  Direction.  —  1.  Find  lines  extending  in  the 
same  direction  and  lines  extending  in  different  directions. 
With  respect  to  direction  what  relations  may  lines  have  ? 

What  is  true  of  the  direction  in  which  the  right  and 
left  edges  of  the  door  extend  ?  Can  you  think  of  them  as 
extending  in  opposite  directions  ?  What  is  true  of  the 
upper  and  lower  edges  of  the  blackboard  ?     Find  other 

1  "When  thinking  of  a  thing's  figure,  we  think  of  the  relation  of 
magnitude  which  its  constituent  parts  bear  to  one  another ;  but  when 
thinking  of  its  size,  we  think  of  the  relation  of  magnitude  which  it, 
as  a  whole,  bears  to  other  wholes."  —  Herbert  Spencer. 


ADVANCED    ARITHMETIC.  139 

edges  in  the  room  that  suggest  the  same  direction  ;  that 
suggest  opposite  directions.  How  many  edges  of  a  cube 
extend  in  the  same  direction  ?  Show  four  edges  of  the 
room  extending  in  the  same  direction.  Try  to  find  three 
groups  of  four  edges  of  the  room  that  extend  in  the  same 
direction.     Do  so  with  a  cube. 


.b 


What  is  true  of  the  direction  in  which  the  lines  ah,  gh, 
cd,  and  om  extend  ? 

Lines  thought  of  as  extending  in  the  same  direction  are 
called  parallel  lines.  Who  has  seen  parallel  roads,  paral- 
lel surfaces,  parallel  sidewalks,  people  walking  in  parallel 
directions  ? 

If  the  parallelism  of  lines  consists  in  their  sameness  of 
direction,  are  Carl  and  Lyle  traveling  in  parallel  directions 
when  one  walks  toward  the  east  and  the  other  toward  the 
west  ?  What  must  one  of  them  do  to  make  the  directions 
parallel  ? 

If  parallel  lines  are  lines  which  extend  in  the  same 
direction,  how  many  pairs  of  parallel  lines  are  suggested 
by  tlie  edges  of  the  black- 
board ? 

2.  If  ah  represents  one 
direction  and  cd  an  opposite 
direction,  are  the  lines  ah  and 
cd  parallel  ?  Are  the  lines 
ah  and  dc  parallel  ?  Why  ?  (Notice  that  the  lines  referred 
to  in  the  last  question  extend  from  a  to  h  and  from  d  to  c. 
The  direction  is  indicated  by  the  order  in  which  the  letters 
are  read.) 


140 


ADVANCED   ARITHMETIC. 


Review. 

Remark.  —  The  perceptions  necessary  to  make  permanent  pos- 
sessions of  the  things  perceived  are  not  repetitions  of  words, 
but  of  ideas,  and  to  get  the  thought  again  and  again  before  the 
pupil  is  possible  only  by  arousing  the  mind  to  activity. 

What  relations  of  magnitude  may  solids  have  ?  surfaces  ? 
lines  ?  What  relations  may  lines  have  with  regard  to  direc- 
tion ?  What  are  parallel  lines  ?  If  a  boy  travels  one  rod 
east,  and  another  one  mile  east,  are  the  directions  parallel  ? 

Angles.  —  1.  What  is  true  of  the  direction  in  which  lines 
ah  and  cd  extend  ?  What  is  true  of  the  directions  in 
which  the  lines  mn  and  eo  extend  ?     Draw  two  lines  so 

that  they  make  a  slight 
difference  in  direction. 
Draw  two  others  so  that 
they  make  a  greater  dif- 
ference in  direction. 

2.  How  must  two 
lines  extend  to  indicate 
a  sameness  in  direc- 
tion ?  to  indicate  a  dif- 
ference in  direction  ? 

3.  Which  is  greater,  the  difference  in  direction  of  the 
lines  7nn  and  eo,  or  of  the  lines  ts  and  tji  ? 

4.  Find  lines  of  a  cube  extending  in  different  directions. 
Find  lines  in  the  room  extending  in  different  directions. 
Find  walls  of  the  room  extending  in  the  same  direction  ;  in 
different  directions. 

5.  How  many  directions  are  required  to  make  a  differ- 
ence in  direction  ?  To  see  a  difference  in  direction  is  to 
see  an  angle.  What  is  an  angle  ?  If  an  angle  is  a  differ- 
ence in  direction,  how  many  directions  are  required  to  make 
an  auijle  ? 


ADVANCED   ARITHMETIC. 


141 


1.  How  many  lines  are  required  to  make  a  sameness  in 
direction  ?  to  make  a  difference  in  direction  ?  The  differ- 
ence in  direction  of  two  lines  lying  in  the  same  plane  is 
called  a  plane  angle. 

2.  Can  any  two  lines  whatsoever  drawn  in  the  same 
plane  and  forming  an  angle  be  made  to  meet  if  they  are 
extended  ?    What  is  the  exception  ? 

3.  Draw  lines  in  opposite  directions.  What  is  an  angle  ? 
Is  there  a  difference  in  the  direction  of  the  lines  just 
drawn  ?  Do  they  form  an 
angle  ?  .If  these  lines  are 
thought  of  as  extending  in 
the  same  direction,  what 
term  is  applied  to  them  ? 

4.  Do  you  see  an  angle 
in  diagram  a?  in  diagram  b  ? 

Which  is  the  greater  angle  ?  Why  is  a  the  greater  angle  ? 
What  makes  one  angle  greater  than  another  ?  Does  it 
make  an  angle  larger  to  lengthen  the  lines  that  form  it  ? 
Can  two  short  lines  make  as  large  an  angle  as  two  long 
lines  ?  Show  this.  Can  the  lines  in  diagram  a  be  pro- 
duced so  that  they  will  meet?    in  diagram  b? 

5.  Draw  lines  in  opposite  directions.   What  is  an  angle  ? 

Do  the  lines  just  drawn 
form  an  angle  ?  Do  lines 
thought  of  as  extending  in 
different  directions  form 
angles  ? 

6.  Can  lines  be  drawn 
on  the  blackboard  which,  if  produced,  will  never  meet? 
Will  the  lines  ab  and  m7i  meet  if  they  are  extended  ? 

7.  Can  all  lines  drawn  on  the  blackboard  and  forming 
angles  be  extended  so  that  they  will  meet  ?  What  is  the 
exception  ? 


142 


ADVANCED    ARITHMETIC. 


If  the  lines  of  an  angle  meet,  the  point  of  meeting  is  called  the  I 
vertex  of  the  angle. 

1.  Draw  an  angle.  Describe  a  circle,  using  the  vertex 
of  the  angle  as  the  center.  Estimate  the  part  of  the  cir- 
cumference of  the  circle  that  is  intercepted  by  the  lines  of 

the  angle.  If  the  circumference 
of  the  circle  is  divided  into  360 
equal  arcs,  or  degrees,  how  many 
of  these  degrees  are  intercepted 
by  the  lines  of  the  angle  ?  Then 
how  many  degrees  measure  the 
difference  in  direction  of  the 
lines  of  the  angle  ? 

2.  Will  the  number  of  degrees 
be  greater  if  the  circle  is  larger  ? . 
less  if  the  circle  is  smaller  ?  Draw  five  angles  of  different 
sizes.  Estimate  the  number  of  degrees  in  each.  Use  a  pro- 
tractor and  measure  the  angles  to  correct  the  estimate  made. 
3.  Draw  an  angle.  Draw  a  larger  angle.  Draw  the 
largest  angle  that  you  can.  ^      c 

In  what  directions  do  the 
lines  of  the  largest  angle 
extend  ? 

1.  Into  how  many  arcs 
each  equal  to  the  arc  ao 
can  the  circumference  of 
this  circle  be  separated  ? 

2.  The  distance  from  a 
to  0  is  60°.  How  many 
60°  in  the  circumference 
of  the  circle  ?     60°  equals  what  part  of  a  circumference  ? 

3.  What  is  the  distance  from  oton?  How  many  30° 
in  the  circumference  of  this  circle  ? 


ADVANCED    ARITHMETIC. 


143 


4.  What  is  the  distance  from  n  to  d?  The  arc  ao 
equals  how  many  15°  ?  The  circumference  of  a  circle 
equals  how  many  15°  ? 

5.  Draw  an  arc  of  60°;  of  30°;  of  15°. 

6.  One  degree  equals  what  part  of  the  circumference 
of  a  circle  ?  1°  equals  60'.  (Bead:  1°  equals  60  minutes 
of  distance.)  60'  equals  what  part  of  the  circumference  of 
a  circle  ?  60'  equals  what  part  of  the  distance  from  n  to 
d  ?  from  0  to  n? 

7.  The  length  of  1°  on  the  equator  is  60  geographical 
miles.     What  is  the  length  of  1'  on  the  equator  ? 

8.  1'  equals  60"  ("  seconds  of  distance).  If  1  mile  is 
the  length  of  1'  on  the  equator,  what  is  the  length  of  1"  on 
the  equator  ? 

9.  If  the  length  of  1'  on  the  parallel  which  forms  the 
north  boundary  of  British  Columbia  is  ^  of  a  geographical 
mile,  what  is  the  length  of  1°  on  the  same  parallel  ?  of  30'  ? 
of  30"?  of  1"?  of  2°? 

10.    1"  equals  what  part  of  1'? 

1'  equals  what  part  of  1°  ? 

1°  equals  what  part  of  the  circumference  of  a  circle  ? 

c 


What  is  true  of  the  direction  in  which  the  lines  ab  and 
ac  extend  ?  The  difference  in  direction  of  the  lines  ab 
and  ac  is  a  straight  angle. 


144  ADVANCED   ARITHMETIC. 

Think  of  the  difference  in  direction  of  lines  extending  in 
opposite  directions. 

What  is  a  straight  angle 

If  a  straight  angle  is  the  difference  in  direction  of  two 
lines  that  extend  in  opposite  directions,  is  the  angle  ond 
a  straight  angle  ?  The  angle  ond  equals  what  part  of  a 
straight  angle  ?     See  p.  143. 

The  angle  07id  is  a  right  angle. 

A  right  angle  equals  what  part  of  a  straight  angle  ? 

What  is  a  right  angle  ? 

Are  the  lines  fp  and  rq  perpendicular  to  each  other  ? 

The  lines  fp  and  rq  form  what  kind  of  an  angle  ? 

The  angle  xme  is  an  acute  angle.  The  acute  angle  xine 
is  less  than  one-half  of  what  angle  ?  It  is  less  than  what 
angle  ? 

What  is  an  acute  angle  ? 

Is  an  acute  angle  less  or  greater  than  a  right  angle  ? 

The  angle  hkl  is  an  obtuse  angle. 

What  is  an  obtuse  angle  ? 

In  what  position  are  the  lines  of  a  right  angle  with 
regard  to  each  other? 

Is  there  any  angle  greater  than  a  right  angle  which  is 
not  an  obtuse  angle  ?     Which  angle  is  it  ? 

1.  Draw  a  right  angle  having  long  lines  and  a  right 
angle  having  short  lines.  Which,  if  either,  is  the  greater  ? 
Why  are  they  equal  ?  What  kinds  of  angles  are  always 
respectively  equal  ? 

2.  What  is  true  of  the  magnitude  of  obtuse  angles? 
of  acute  angles  ?  Are  obtuse  angles  always  equal  ?  acute 
angles  ? 

Note.  —  Acute  and  obtuse  angles  are  also  called  oblique  angles, 
and  the  lines  which  form  them  oblique  lines. 


ADVANCED   ARITHMETIC.  145 

3.    Give  names  of   the  angles   in  the  diagram.     How 
many  are  right  ?    straight  ?   acute  ?    obtuse  ?    oblique  ? 


4.  Find  two  right  angles.     Find  the  two  pairs  of  per- 
pendicular lines. 

5.  Find  the  three  oblique  angles.     How  many  of  the 
oblique  angles  are  acute  ? 

6.  Find  three  pairs  of  oblique  lines.     The  oblique  lines 
are  the  lines  of  which  angles  ? 

7.  Find  edges  of  solids  forming  right  angles  ;  acute ; 
obtuse. 

8.  Do   you   know    of   any  streets    that   form   oblique 
angles  ? 

9.  How  many  angles   are    made    by  two   intersecting 
streets  or  by  two  intersecting  lines  ? 

10.  If  one  of  the  four  angles  formed  by  two  intersecting 
lines  is  a  right  angle,  what  is  true  of  the  magnitude  of  the 
other  three  ?  if  one  is  acute  ?  if  one  is  obtuse  ? 

fv 

To  Draw  Perpendicular  Lines.  —  It  will  aid  greatly  in  making 
boxes,  cubes,  and  other  solids  accurately  and  easily,  to  learn  to 
draw  perpendicular  lines  and  to  handle  the  ruler  and  dividers 
readily. 

The  pupils  should  first  see  the  drawings  as  wholes.  Afterward 
they  may  observe  the  teacher  draw  and  learn  a  method  of  proceed- 
ing by  seeing  and  doing,  not  by  following  step  by  step  a  formal 


146 


ADVANCED   ARITHMETIC. 


direction  such  as,  "  Let  AB  be  a  given  line  and  D  a  point  within 
it.     From  D  as  center,"  etc. 

Growth  in  the  power  to  express  orally  should  increase  as  per- 
ceptions and  power  to  express  by  hand  increase.  But  there  is 
danger  of  confusing  the  pupil  by  pushing  the  means  into  the  fore- 
ground through  verbal  description.     When  he  has  a  mental  whole 

which  he  wishes  to  construct 
F  and  sees  the  teacher  make  it 

many  times  easily  and  quickly, 
"^<E  he  readily  gains  similar  con- 

structive power,  without  dwell- 
ing on  the  mechanism. 


Practice  drawing  perpen- 
dicular lines.  Write  a 
direction  for  drawing  per- 
pendicular lines. 

1.  Can  you  draw  right  angles  of  different  sizes  ?   Try  it. 

2.  Can  you  draw  straight  angles  of  different  sizes  ? 
Try  it. 

3.  How  many  acute  angles  of  different  sizes  may  be 
drawn  ? 

4.  How  many  obtuse  angles  of  different  sizes  may  be 
drawn  ? 

5.  What  is  the  least  possible  number  of  acute  angles 
that  may  form  one  right  angle  ? 

6.  What  is  the  least  possible  number  of  acute  angles 
that  may  form  an  obtuse  angle  ? 

7.  What  is  the  least  possible  number  of  acute  angles 
that  may  form  a  straight  angle? 

8.  An  obtuse  angle  and,  at  least,  how  many  acute 
angles  equal  a  straight  angle  ? 

9.  The  sum  of  a  right  angle  and  what  equals  an  obtuse 
angle  ?  What  equals  the  difference  between  a  right  angle 
and  an  obtuse  angle  ? 


ADVANCED    ARITHMETIC.  147 

10.  The  sum  of  a  right  angle  and  what  equals  a  straight 
angle  ?  What  equals  the  difference  between  a  right  angle 
and  a  straight  angle  ? 

Review.  — What  is  space  ?  What  is  a  solid  ?  What  is 
the  limit  of  a  solid  ?  What  is  a  straight  line  ?  What  is  a 
point  ?  What  is  a  plane  ?  What  are  parallel  lines  ?  What 
is  an  angle  ?  What  relations  of  magnitude  may  solids 
have  ?  surfaces  ?  lines  ?  What  relations  of  direction  may- 
lines  have  ?  What  is  a  plane  angle  ?  What  is  the  vertex 
of  an  angle  ?  Define  the  different  kinds  of  angles.  What 
lines  are  perpendicular  ?  What  general  name  is  given  to 
acute  and  obtuse  angles  ?     What  lines  are  oblique  ? 

To  Teacher.  —  The  reviews  are  not  to  be  a  mere  matter  of 
question  and  answer.  It  is  not  expected  that  the  student  will  at 
this  stage  grasp  the  full  significance  of  these  ideas  ;  the  developing 
mind  must  have  time  and  increasing  experience  to  give  ideas 
expansion.  But  the  opportunity  for  seeing  things  in  new  lights, 
for  making  the  needful  associations,  should  be  freely  afforded. 
Attempts  to  apply  the  adult  standard  will  defeat  growth  into 
higher  and  more  complete  ideas.  There  can  be  no  real  review 
and  no  real  progress  without  renewing  interest  from  time  to  time, 
and  thus  gradually  unifying  and  enlarging  ideas. 

1.  What  is  a  plane  ? 

2.  What  is  a  polygon  ? 

3.  In  what  respect  are  the  polygons  alike  ? 

4.  What  kind  of  a  figure  is  a  polygon  ?     By  what  are 
polygons  bounded  ? 

5.  What  is  a  polygon  ?     A  polygon  is  a  plane  bounded 
by  straight  lines. 

6.  What  is  the  least  number  of  sides  a  polygon  may 
have  ?  the  greatest  number  ? 

7.  What  is    a  triangle  ?     A  triangle  is   a  polygon  of 
three  sides. 


148 


ADVANCED   ARITHMETIC. 


POLYGONS. 


8.  What  is  a  quadrilateral  ?   a  peMtagori  ?    a  hexagon  ? 
a  heptagon  ?  an  octagon  ? 

9.  What  does  the  word  ^'■polygon-''  mean  ? 

10.    Find  polygons  in  the  room  and  tell  how  many  sides 


ADVANCED    AHITHMETIC. 


149 


they  have.  Ex.  The  surface  of  the  blackboard  is  a  poly- 
gon of  four  sides.  The  base  of  this  hexagonal  prism  is  a 
polygon  of  six  sides. 


Quadrilaterals.  —  1.  Draw  quadrilaterals.  Draw  equi- 
lateral quadrilaterals.  Draw  an  oblique-angled  quadri- 
lateral.    Draw  a  right-angled  one. 

2.  How  many  of  these  polygons  are  quadrilaterals  ? 
How  many  are  equilaterals  ? 

3.  How  many  are  right-angled?  How  many  are 
oblique-angled  ? 

4.  In  what  are  a,  6,  c,  and  d  alike  ?  What  is  true  of 
the  opposite  sides  of  a,  h,  c,  and  d  ?  What  common  name 
do  a,  h,  c,  and  d  have  that  e  and  /  do  not  have  ? 

5.  How  maftiy  of  these  polygons  are  parallelograms  ? 

6.  In  what  are  a  and  h  alike  ?    In  what  do  they  differ  ? 

7.  What  common  name  do  a  and  b  have  that  the  other 
parallelograms  do  not  have  ? 

8.  In  what  are  c  and  d  alike  ?  In  what  do  they  differ  ? 
What  common  name  do  c  and  d  have  that  a  and  h  do  not 
h^ve? 


150  ADVANCED   ARITHMETIC. 

9.    Are   all   quadrilaterals    parallelograms?      Are   all 
parallelograms  quadrilaterals  ? 

10.  Are  all  squares  rectangles  ?  Are  all  rectangles 
squares  ? 

11.  Which  quadrilateral  has  only  one  pair  of  parallel 
lines  ?     Which  has  no  parallel  lines  ? 

12.  How  many  kinds  or  classes  of  quadrilaterals  are 
there  ? 

13.  Cut  two  of  each  kind,  making  each  pair  as  unlike 
as  possible.  In  which  pair  is  there  the  least  variation  in 
shape  ?  Is  there  not  one  pair  in  which  there  is  no  varia- 
tion in  the  angles  nor  in  the  relative  length  of  the  lines  ? 
Which  is  it  ?  In  one  pair  what  is  the  only  variation  that 
can  be  made  ?  Why  is  there  less  variation  in  the  shape  of 
the  quadrilaterals  having  right  angles  than  in  those  having 
oblique  angles  ? 

14.  What  is  wrong  with  this  definition  of  a  square  :  A 
square  is  a  right-angled  parallelogram  ? 

15.  What  is  wrong  with  this  :  A  square  is  an  equilateral 
parallelogram  ?  Make  a  definition  of  a  square  that  will 
not  include  any  other  parallelogram. 

16.  Write  in  the  fewest  words  possible  such  a  description 
of  each  quadrilateral  that  the  person  reading  it  can  select 
the  quadrilateral  described.  Ex.  Find  or  draw  an  equi- 
lateral oblique-angled  parallelogram. 

17.  Is  there  more  than  one  kind  of  quadrilateral  that  is 
both  equilateral  and  oblique-angled  ? 

18.  ^  Write  a  definition  of  each  of  the  following  : 

1  "The  axioms  require  not  to  be  granted,  but  to  be  seen.  If  any 
one  were  to  assent  to  them  without  seeing  them  to  be  true,  his  assent 
would  be  of  no  avail  for  purposes  of  reasoning.  .  .  .  Supposing  we 
could  deduce  our  reasoning  from  definitions  alone,  it  must  be  allowed, 
I  think,  that  still  our  geometrical  propositions  would  properly  depend, 
not  on  the  definitions,  but  on  the  act  of  the  mind  by  which  we  fix 
upon  such  definitions." — Dr.  Whewell.  / 


I 


ADVANCED    ARITHMETIC. 


151 


A  polygon,  a  quadrilateral,  a  parallelogram,  a  rectangle, 
a  square,  a  rhomboid,  a  rhombus,  a  trapezoid,  a  trapezium. 

Picture  mentally  the  different  quadrilaterals. 

A  quadrilateral  is  a  polygon  of  four  sides. 

A parallelogravi  is  a  quadrilateral  whose  sides  are  parallel. 

A  rectangle  is  a  right-angled  parallelogram. 

A  square  is  an  equilateral  rectangle. 

A  rliomhoid  is  an  oblique-angled  parallelogram. 

A  rhombus  is  an  equilateral  rhomboid. 

A  trapezoid  is  a  quadrilateral  having  only  one  pair  of 
parallel  sides. 

A  trapezium  is  a  quadrilateral  having  no  sides  parallel. 

What  is  a  square  ?  a  rectangle  ?  a  parallelogram  ?  a 
quadrilateral  ?   a  polygon  ? 

Review  from  the  beginning. 


Finding  Forms  made  by  Folding.  —  Give  each  pupil  a 
four-inch  square.     Place  it  for  folding. 

Fold  the  square  so  that  the  right-hand  point  in  front 
will  coincide  with  the  left-hand  point  at  the  back. 

Crease.  Open  the  paper.  Fold  the  square  so  that  the 
left-hand  point  in  front  will  fall  upon  the  right-hand  point 
at  the  back. 

Crease.  Open  the  paper.  Fold 
the  paper  so  that  the  front  edge 
will  fall  upon  the  diagonal  which 
was  creased  first. 

Crease.  Fold  the  left-hand  edge 
so  that  it  will  coincide  with  the 
same  diagonal. 

Crease.    Fold  the  paper  so  that 
the  right-hand  point  at  the  back  will  fall  upon  a  diagonal, 
and  so  that  an  isosceles  triangle  will  be  formed. 

Crease  paper  carefully.     Unfold  the  paper  so  that  you 


152 


ADVANCED   ARITHMETIC. 


will  have  the  square  again.     Observe  the  forms  made  by 
the  creased  lines. 


1.  Observe  the  figure.     How  many  triangles  each  hav- 
ing a  right  angle  can  you  find  ? 

2.  How  many  triangles  having  two  sides  equal  can  you 
find? 

3.  How  many  triangles  can  you  find  having  no  two 
sides  equal? 

4.  How  many  trapezoids  can  you  find  ? 

5.  How  many  different  figures  have  you  found  in  the 
square  ? 

To  Teacher Draw  the  square  several   times  in  presence  of 

pupils.     See  note,  page  145.' 


Make  cubes  and  other  solids.^ 

1  Making  some  of  the  simple  solids  at  this  time  prepares  for  the 
development  of  the  more  complex  solids  when  the  study  of  the  solids 
is  begun.    The  work  of  making  the  solids  can  be  done  at  home.    The 


ADVANCED    ARITHMETIC. 


153 


Suggestion.  —  Draw  on  thick,  tough  cardboard  the  diagram 
shown  below.  Cut  the  diagram  out  of  the  cardboard  and  cut 
half  through  the  edges  of  the  surfaces  that  are  joined.  Then  fold 
into  the  solid.  Fasten  the  adjacent  surfaces  of  the  solid  by  means 
of  mucilage  or  paste. 


Areas  of  Parallelograms.  —  Show  the  line  on  which  each 
of  the  parallelograms  rests  and  the  line  opposite,  i.e.,  show 
the  bases  of  the  parallelograms. 


Show  the  bases  of  the  blackboard. 
Find  other  parallelograms  and  show  their  bases. 
The  altitude  or  height  of  a  parallelogram  is  the  perpen- 
dicular distance    between  what  ?     What   is   true   of   the 

first  work  handed  in  by  the  pupils  will  probably  be  soiled  and  poorly 
put  together.  Mistakes  in  the  diagram  will  be  manifest  in  imperfect 
solids.  Continued  practice'  is  needed  to  give  muscular  control  and 
the  habit  of  ready  and  effective  action.  Such  action  is  the  product 
of  growth.  The  particular  thing  made  is  of  consequence  only  as  it 
affects  the  condition,  —  the  development  of  the  individual.  A  display 
of  finely  finished  forms  is  not  necessarily  evidence  of  growth. 


154  ADVANCED   ARITHMETIC. 

comparative  altitude  of  the  parallelograms  above  ?  Draw 
oblique-angled  parallelograms  on  the  blackboard  and  meas- 
ure their  altitude. 

Review  pp.  279  to  282,  "Elementary  Arithmetic." 

Areas  of  Rectangles.  —  1.  Draw  a  rectangle  5  in.  long 
and  3  in.  wide. 

What  is  the  number  of  sq.  in.  in  a  rectangle  5  in.  long 
and  1  in.  wide  ? 

What,  then,  is  the  number  in  a  rectangle  5  in.  long  and 
3  in.  wide  ? 

2.  What  is  the  area  of  a  rectangle  5  in.  long  and  4^ 
in.  wide? 

9  '5 

— ^  =  the  number  of  sq.  in.  in  the  area. 

What  is  the  number  of  sq.  in.  in  a  rectangle  5  in.  long 
and  \  in.  wide  ?  What,  then,  equals  the  number  in  a  rect- 
angle 5  in.  long  and  |  in.  or  4^  in.  wide  ? 

3.  Find  the  area  of  a  rectangle  6f  ft.  long  and  3f  ft.  wide. 

'^     =  the  number  of  sq.  ft.  in  the  area. 

What  is  the  number  of  sq.  ft.  in  a  rectangle  6f  ft.  long 
and  1  ft.  wide  ?  What,  then,  equals  the  number  of  sq.  ft. 
in  a  rectangle  6f  ft.  long  and  }  ft.  wide  ?  What,  then, 
equals  the  number  of  sq.  ft.  in  a  rectangle  6f  ft.  long  and 
2A  or  3f  ft.  wide  ? 

4.  What  is  the  number  of  sq.  in.  in  the  top  of  a  writing 
table  2J  ft.  long  and  1^  ft.  wide  ? 

5.  What  is  the  area  of  a  window  pane  whose  width  is 
2|  ft.  and  whose  altitude  is  2^%  ft.  ? 

6.  Find  the  number  of  eq.  yd.  in  the  floor  of  the  school- 
room. 

7.  A  garden  is  8^  rd.  long  and  5^  rd.  wide.     What  is 
its  area? 


ADVANCED   ARITHMETIC.  155 

8.  How  many  sq.  in.  in  a  sheet  of  paper  8"  X  13"? 

9.  How  many  sq.  ft.  in  the  upper  surface  of  6  paving 
stones  f  ft.  long  and  ^  ft.  wide  ? 

10.  What  is  the  number  of  sq.  yd.  in  the  ceiling  of  a 
room  18^  ft.  long  and  16  ft.  wide  ? 

If  the  room  is  12  ft.  high,  what  is  the  number  of  sq.  yd. 
in  the  walls  of  the  room  ? 

What  is  the  ratio  of  the  number  of  sq.  yd.  to  the  number 
of  sq.  ft.  ? 

11.  If  a  is  the  number  of  ft.  in  the  length  of  a  room,  h 
the  number  in  its  width,  and  c  the  number  in  its  altitude, 
what  equals  the  number  of  sq.  yd.  in  the  walls  of  the 
room  ?   in  the  ceiling  ? 

12.  How  much  pasteboard  is  required  to  make  a  box 
3"  X  8"  X  12"  ?     (No  allowance  for  laps  or  cover.) 

How  much  is  required  for  the  cover,  allowing  it  to  turn 
down  1"  all  around? 

13.  How  much  surface  is  covered  in  painting  a  rectangu- 
lar prism  3"X7"X11"? 

14.  How  many  sq.  in.  in  the  lateral  surface  of  an  equi- 
lateral triangular  prism  whose  base  is  6"  on  each  edge  and 
whose  height  is  12"? 

15.  How  many  sq.  in.  in  the  lateral  surface  of  a  hex- 
agonal prism  whose  base  is  4"  on  each  edge  and  whose 
height  is  8"? 

16.  ^  What  is  the  ratio  of  the  number  of  sq.  ft.  in  a  floor 
10  ft.  X  12  ft.  to  the  number  of  sq.  ft.  in  a  board  of  flooring 
12  ft.  long  and 4  in.  wide? 

Then  how  many  such  boards  are  required  for  tlie 
floor  ? 

17.  A  strip  of  land  containing  2  acres  is  ^  of  a  mile 
long.  How  wide  is  it  ?  (See  '^  Elementary  Arithmetic," 
pp.  253-257.) 

1  See  number  ratios,  pp.  251-253,  "Elementary  Arithmetic." 


156 


ADVANCED   ARITHMETIC. 


18.  What  equals  the  number  of  sq.  ft.  of  surface  covered 
in  oiling  one  side  of  a  door  6'  10"  high  by  2'  4"  wide  ? 

7 '41 

- — —  =  number  of  sq.  ft. 
o  ■  D 

19.  How  many  sq.  yd.  of  plastering  in  the  walls  of  a 
room  15  ft.  long  X  12  ft.  wide  X  10  ft.  high  ?  How  many 
sq.  yd.  in  the  ceiling  ? 

Give  the  dimensions  of  a  rectangle  equal  to  the  surface 
of  the  four  walls.  How  many  sq.  ft.  in  this  rectangle? 
What  is  the  ratio  of  the  number  of  sq.  yd.  to  the  number 
of  sq.  ft.  ? 

20.  What  is  the  ratio  of  a  rectangle  S^"  long  and  i" 
wide  to  a  sq.  in.  ? 

21.  What  is  the  ratio  of  a  rectangle  2J-"  long  and  J-" 
wide  to  a  sq.  in.  ?  What  is  the  ratio  of  a  rectangle  Sy 
long  and  ^"  wide  to  a  sq.  in.  ? 

22.  At  $x  a  sq.  rd.,  what  equals  the  cost  of  a  square  field 
whose  perimeter  is  y  rd.? 

23.  If  a  equals  the  number  of  ft.  in  the  length  of  a 
room  and  b  the  number  of  sq.  ft.  in  the  area,  what  equals 
the  number  of  ft.  in  the  width  ? 

24.  How  many  yd.  of  matting  f  yd.  wide  are  needed  to 

cover  a  floor  21  ft.  square  ? 

25.  At  $1-  a  sq.  yd.,  what 
is  the  cost  of  painting  a  roof 
37^  ft.  long  and  24  ft.  wide? 

26.  At  $i  a  sq.  yd.,  what 
is  the  cost  of  painting  both 
sides  of  a  board  fence  110.4 
ft.  long  and  5.25  ft.  high  ? 


To  Draw  an  Equilateral  Tri- 
angle. —  Let  AB  be  a  side  of  the  triangle. 

From  A  and  B  as  centers,  and  with  a  radius  equal  to 


ADVANCED   ARITHMETIC. 


157 


AB,  describe  arcs  intersecting  at  D.     Join  DA  and  DB. 
Practice  drawing  equilateral  triangles.     See  p.  145. 


To    make  a  triangular   pyramid   of  which  each  edge  shall 
.    be  3  in. 


Construct  an  equilateral  triangle  whose  edge  is  6  in. 
Separate  this  equilateral  triangle  into  four  equal  equi- 
lateral triangles. 


158 


ADVANCED   ARITHMETIC. 


ADVANCED   ARITHMETIC. 


159 


1.  Find  all  the  ratios  you  can.^     See  p.  158. 

2.  Find  equal  units. 

3.  Find  the  ratio  2  ;  the  ratio  J. 

4.  What  is  the  ratio  of  a  to  c  ?  of  o  to  c  ?  of  a  to  o  ? 

5.  What  is  the  ratio  oi  ato  k?  oi  do  k?  of  o  to  k? 

6.  What  is  the  ratio  of  m  to  s  ?  of  ??i  to  r  ?  of  m  to  u  ? 
of  m  to  ^  ? 

7.  What  is  the  ratio  of  a  to  5  ?  of  Z  to  s  ?  of  Z  to  a  ? 

8.  What  is  the  ratio  of  h  to  the  square  k?  ofhtor? 

9.  What  is  the  ratio  of  6  to  ^  ?  of  A  to  A;  ?  of  ^^  to  A  ? 

10.  h  equals  ^of  k;  b  equals  ^oi  k-,  therefore  h  and  b  are 
equal.  Things  that  are  equal  to  the  same  thing  are  equal 
to  each  other. 

11.  Prove  that  b  and  s  are  equal. 


Areas  of  Rhomboids. — 1.  Cut  a  rectangle  having  a  base 
of  4  in.  and  an  altitude  of  2  in.  Cut  a  rhomboid  having  a 
base  and  altitude  equal  to  those  of  the  rectangle.  What  is 
true  of  the  area  of  the  two  parallelograms  ? 

2.  What  is  the  area  of  a  rectangle  whose  base  is  4  in. 
and  altitude  2  in.  ?  What  is  the  area  of  a  rhomboid  whose 
base  is  4  in.  and  altitude  2  in.  ? 

3.  If  17  is  the  number  of  ft.  in  the  base  of  a  rectangle 
and  3  is  the  number  of  ft.  in  its  altitude,  what  equals  the 
number  of  sq.  ft.  in  its  area  ?  If  17  is  the  number  of  ft. 
in  the  base  of  a  rhomboid  and  3  is  the  number  of  ft.  in 
its  altitude,  what  equals  the  number  of  sq.  ft.  in  its  area  ? 

1  If  a  portion  of  several  days  were  given  to  the  study  of  the  rela- 
tions in  the  above  diagram,  it  would  not  result  in  a  loss  of  time  to  the 
pupils. 


160  ADVANCED    ARITHMETIC. 

4.  What  is  the  area  of  a  rectangle  whose  base  is  11  in. 
and  altitude  5  in  ?  What  is  the  area  of  a  rhomboid  whose 
base  is  11  in.  and  altitude  5  in.  ? 

5.  How  many  acres  in  a  rectangle  80  rd.  long  and  14 
rd.  wide  ? 

=  the  number  of  what  ? 
loO 

How  many  acres  in  a  field  in  the  shape  of  a  rhomboid  if 

its  base  is  80  rd.  and  its  altitude  14  rd.  ? 

6.  Cut  rhomboids  of  different  dimensions.  Measure 
base  and  altitude  and  find  area  of  each. 

7.  What  are  the  names  of  the  two  kinds  of  parallelo- 
grams ? 

8.  Draw  different  kinds  of  parallelograms  on  the  black- 
board. Measure  base  and  altitude  of  each  and  find  the 
number  of  units  in  the  area  of  each. 

1.  What  is  the  area  of  a  square  a  in.  in  length  ? 

2.  If  a  is  the  number  of  rd.  in  the  length  of  a  rectangular 
piece  of  land  and  h  is  the  number  of  rd.  in  the  width,  what 
equals  the  number  of  sq.  rd.  in  the  area? 

3.  What  is  the  area  of  a  rhombus  whose  length  is  a  ft. 
and  whose  altitude  is  h  ft.  ?  What  is  the  perimeter  of  the 
rhombus  ? 

4.  What  is  the  area  of  a  rhomboid  a  ft.  long  and  b  ft. 
wide  ? 

5.  A  rectangular  piece  of  land  1  mile  long  and  ^  mile 
wide  contains  how  many  acres  ? 

6.  How  many  rd.  of  fence  will  enclose  a  farm  1  mile 
square  ? 

7.  How  much  additional  fence  will  divide  it  into  four 
equal  square  fields  ? 

8.  How  many  sq.  yd.  of  cloth  will  it  take  to  cover  the 
tops  of  all  the  desks  in  this  room  ? 


ADVANCED    ARITHMETIC.  161 

9.    a  is  the  area  of  a  rectangle  and  h  its  altitude.    What 
is  its  base  ? 

10.  a  is  the  area  of  a  rhombus  and  h  its  altitude.  What 
is  its  base  ? 

11.  a  is  the  area  of  a  rectangle  and  h  its  base.  What  is 
its  altitude  ? 

12.  a  is  the  area  of  a  rhomboid  and  h  its  base.  What  is 
its  altitude  ? 

13.  a  is  the  area  of  a  parallelogram  and  ^  its  altitude. 
What  is  its  base  ? 

14.  A  piece  of  land  40  rd.  long  and  4  rd.  wide  contains 
an  acre ;  how  wide  is  a  piece  having  the  same  area  if  it  is 
only  I  as  long  ? 

15.  An  open  court  contains  40  sq.  yd.  How  many  stones 
9  in.  sq.  will  be  required  to  pave  it  ? 

16.  How  many  sq.  in.  in  the  entire  surface  of  a  brick 
9  in.  long,  6  in.  wide,  and  4  in.  thick  ? 

17.  What  is  the  length  of  a  parallelogram  whose  width 
is  120  rd.  and  whose  area  is  60  A.  ? 

18.  What  is  the  width  of  a  rectangular  field  whose 
length  is  16|  rd.  and  whose  area  is  12^  A.  ? 

What  would  the  width  be  if  the  area  were  16§  sq.  rd.  ? 

19.  The  perimeter  of  a  square  field  equals  160  rd. 
What  is  the  perimeter  of  another  square  field  having  four 
times  the  area  of  the  first  ? 

20.  How  many  steps  of  2  ft.  6  in.  each  will  a  man  take 
in  walking  around  a  field  45  rd.  square  ? 

21.  How  much  will  it  cost  to  lay  a  pavement  36  ft.  long 
and  9  ft.  6  in.  wide  at  40/  a  sq.  yd.  ? 

22.  How  many  sq.  ft.  in  a  city  lot  62  J  ft.  front  and  208 
ft.  deep  ? 

23.  How  many  tiles  8  in.  square  will  lay  a  floor  48  ft. 
by  10  ft.  ? 

24.  At  $2  a  rd.,  how  much  less  will  it  cost  to  fence  a 


162 


ADVANCED    ARITHMETIC. 


piece  of  land  80  rd.  square  than  a  rectangular  field  twice 
as  long  and  one-half  as  wide  ? 

25.  What  is  the  area  of  a  field  in  the  form  of  a  parallelo- 
gram whose  length  is  30  rd.  and  the  perpendicular  distance 
between  whose  sides  is  24  rd.  ? 

26.  Find  the  area  of  the  blackboard  surface  in  the  room ; 
the  area  of  the  ceiling. 

27.  A  rectangular  lawn  18  ft.  long  and  16  ft.  wide  has 
a  gravel  walk  extending  around  it  on  the  outside.  If  the 
walk  is  3  ft.  wide,  how  many  sq.  ft.  in  it  ? 


ADVANCED    ARITHMETIC. 


163 


1.  Find  all  the  ratios  you  can.    See  p.  162.    Find  equals. 
Find  the  ratio  2. 

2.  What  is  true  of  tsuidu?     Why  ? 

3.  What  is  the  ratio  of  the  sum  of  t  and  utop? 


Areas  of  Triangles.  —  1.  Measure  the  base  and  altitude 
(height)  of  each  triangle. 

2.  What  is  true  of  the  bases  of  triangles  M  and  0  ?  of 
their  altitudes  ? 

3.  What  is  the  base  of  a  triangle  ?  What  is  the  alti- 
tude of  a  triangle  ? 

The  base  of  a  triangle  is  the  side  measured  in  finding  its  area. 
The  vertex  of  a  triangle  is  the  vertex  of  the  angle  opposite  the  base. 
The  altitude  of  a  triangle  is  the  perpendicular  distance  from  its 
verte:^  to  its  base  or  its  base  produced. 

4.  What  must  be  done  before  the  altitude  of  triangle  0 
can  be  measured  ? 

5.  Draw  several  triangles.  Use  as  a  base  a  side  that 
must  be  produced  to  find  the  altitude.  What  is  the  altitude 
of  each  ? 

6.  Measure  the  base  and  altitude  of  each  of  the  paral- 
lelograms on  p.  164. 

What  is  true  of  the  bases  and  altitudes  of  the  parallel- 
ograms ? 

What  is  true  of  the  areas  of  the  parallelograms  ? 

7.  Show  lines  that  are  equal.  Show  parallelograms  that 
are  equivalent.     Draw  two  equal  parallelograms.     Draw 


164 


ADVANCED   ARITHMETIC. 


two  equivalent  parallelograms.  What  is  the  distinction 
between  equal  and  equivalent?  Which  of  the  figures  in 
the  diagram  are  equal  and  which  equivalent  ? 

8.   What  can  be  said  of  the  bases  of  the  triangles  A^  B, 


C,  and  B?  ot  their  altitudes  ?  of  their  areas  ?     Each  tri- 
angle equals  what  part  of  a  parallelogram  ? 

9.    Cut  two  equal  triangles.     Can  these  two  triangles  be 
arranged  so  that  they  will  make  a  parallelogram  ? 

10.  Can  a  triangle  be  made  that  is  not  equal  to  one-half 
of  a  parallelogram  having  an  equal  base  and  altitude  ? 
Try  it. 

11.  Can  you  cut  or  draw  a  triangle  which  is  not  equal  to 
half  a  parallelogram  having  an  equal  base  and  altitude  ? 

12.  The  largest  triangle  that  can  be  cut  out  of  a  square 
equals  what  part  of  the  square  ? 

13.  The  largest  triangle  that  can  be  cut  out  of  a  paral- 
lelogram equals  what  part  of  the  parallelogram  ? 

14.  What  is  true  of  triangles  having  equal  bases  and 
equal  altitudes  ?  Why  do  you  think  they  are  equiva- 
lent ? 

15.  Draw  a  triangle.  Draw  a  rectangle  having  an  equal 
base  and  an  equivalent  area. 

16.  Draw  a  rectangle.  Draw  a  triangle  having  an  equal 
base  and  half  the  area. 


ADVANCED    ARITHMETIC.  165 

17.  What  is  the  altitude  of  a  rectangle  9  in.  long  that  is 
equivalent  to  a  triangle  whose  base  is  9  in.  and  altitude 
6  in.  ? 

18.  Draw  a  number  of  triangles  of  different  sizes  and 
shapes.     Draw  lines  that  are  equal,  respectively,  to  the 


bases  of  the  triangles.     On  these  lines  as  bases  construct 
rectangles  that  are  equivalent  respectively  to  the  triangles. 
19.    Image  rectangles  equivalent  to  triangles  whose  bases 
and  altitudes  are  as  follows  : 


Base. 

Altitude 

12 

6 

10 

7 

14 

8 

12^ 

3 

20.  Find  the  areas  of  each  of  the  above  triangles. 

21.  Cut,  and  find  areas  of  triangles. 

22.  What  is  the  area  of  a  triangular  field  whose  base  is 
48  rd.  and  whose  altitude  is  28  rd.  ? 

23.  The  area  of  a  triangle  is  bOt  sq.  in. ;  its  base  is  8". 
What  is  its  altitude  ? 

24.  A  triangular  field  contains  36  A. ;  its  base  is  J  of  a 
mile.  What  is  its  altitude  ? 

1.  Find  ratios.     See  p.  166. 

2.  Find  umts  that  are  equal. 


166 


ADVANCED    ARITHMETIC. 


3.  What  is  the  ratio  oi  A^  to  B^  ?  of  ?i  to  tlie  sum  of  o 
and  m  ? 

4.  What  is  the  ratio  of  n'  to  the  sum  of  o'  and  m'  ? 

5.  Construct  equal  squares.  Separate  into  squares  and 
triangles,  as  in  ^^  and  B\  Can  you  construct  a  square  on 
the  long  edge  of  a  right  triangle  that  is  not  equal  to  the 


o 

b'^ 

m 

A2  B2 

sum  of  the  squares  on  the  other  two  sides  ?  'Is  the  square 
of  the  hypothenuse  of  a  right  triangle  equal  to  the  sum 
of  the  squares  on  the  other  two  sides  ? 

6.  In  (7,  what  equals  the  area  of  ?i'?     If  the  area  of 
m'  and  of  n*  are  given,  how  can  the  area  of  o'  be  found  ? 

7.  If  the  length  of  o'  and  of  m'  are  given,  how  can  the 
area  of  v]  be  found  ? 

8.  If  the  hypothenuse  of /and  the  side  of  o'  are  given, 
how  may  the  area  of  m'  be  found  ? 


ADVANCED    ARITHMETIC. 


167 


Areas  of  Polygons.  —  Draw  polygons  on  the  blackboard 
similar  to  the  following  : 


22  23  24 

1.   Discover  different  ways  of  finding  the  areas  of  each. 


168 


ADVANCED   ARITHMETIC. 


2.    Explain  method  of  finding  areas,  then  measure  the 
polygons  and  find  areas. 


1.  Find  all  the  ratios  you  can. 

2.  What  is  the  ratio  oia  to  d?  of  d  to  o?  of  a  to  o? 
of  c  to  0?  of  b  to  0? 

3.  What  is  the  ratio  of  A  to  ^  ?  of  h  to  n?  of  m  to  n? 
of  s  to  n?  of  r  to  n? 

4.  How  may  the  area  of  n  be  found  ?  of  ^  ?  of  h? 

5.  Draw  a  line  equal  to  the  sum  of  the  bases  in  diagram  t. 


\ 


ADVANCED    ARITHMETIC.  169 

On  this  as  a  base  construct  a  rectangle  equal  to  the  sum  of 
the  five  triangles. 

6.  The  base  of  one  triangle  is  9",  of  another  13";  the 
altitude  of  each  is  8".  How  can  a  rectangle  equivalent  to 
the  sum  of  the  triangles  be  drawn  ? 

7.  Give  a  short  method  of  finding  the  area  of  a  number 
of  triangles  having  equal  altitudes. 

Triangles.  — 1.  Draw  a  triangle  a  in.  long  and  b  in.  high  ; 
draw  a  rectangle  a  in.  long  equivalent  to  the  triangle. 

2.  Draw  a  rectangle  equivalent  to  the  area  of  two  tri- 
angles whose  bases  are  each  a  in.  and  wliose  altitudes  are 
each  b  in. 

3.  If  a  is  the  base  of  a  triangle  and  b  is  its  altitude, 
what  equals  its  area  ? 

4.  If  a  is  the  area  of  a  triangle  and  b  its  altitude,  what 
equals  its  base  ? 

5.  If  a  is  the  area  of  a  triangle  and  b  its  base,  what 
equals  its  altitude? 

6.  Draw  a  triangle  2  in.  long  and  1 J  in.  high ;  draw  a 
rectangle  2  in.  long  equivalent  to  the  triangle. 

7.  Draw  a  rectangle  equivalent  to  the  area  of  three  tri- 
angles whose  bases  are  2  in.,  3  in.,  and  4  in.,  respectively, 
the  altitude  of  each  being  4  in. 

8.  Divide  a  triangle  into  three  equal  parts  and  state 
why  they  are  equal. 

9.  Prove  that  if  equals  be  taken  from  equals,  equals 
remain.  ' 

10.  What  is  the  ratio  of  a  rectangle  a  in.  long  and  b  in. 
high  to  the  sum  of  two  triangles  each  a  in.  long  and  b  in. 
high  ? 

11.  What  is  the  ratio  of  a  triangle  whose  base  is  x  in. 
and  altitude  y  in.  to  a  rectangle  whose  base  is  x  in.  and 
altitude  y  in.? 


170  ADVANCED   ARITHMETIC. 

12.  What  is  the  ratio  of  a  rectangle  whose  base  is  12  in. 
and  altitude  8  in.  to  a  triangle  whose  base  is  12  in.  and 
altitude  16  in.? 

13.  What  may  be  the  dimensions  of  a  rectangle  equiva- 
lent in  area  to  a  triangle  whose  base  is  12  in.  and  altitude 
Gin.? 

14.  What  may  be  the  dimensions  of  a  triangle  (base  and 
altitude)  equivalent  in  area  to  a  rectangle  whose  base  is  10 
in.  and  altitude  4  in.  ? 

15.  What  may  be  the  dimensions  of  a  rectangle  equiva- 
lent in  area  to  three  triangles  whose  bases  are  4  in.,  5  in., 
and  6  in.,  respectively,  the  altitude  of  each  being  8  in.  ? 

16.  What  are  the  dimensions  of  a  square  whose  perim- 
eter is  12  in.  ? 

17.  What  equals  the  area  of  a  rhombus  whose  perimeter 
is  a  and  altitude  h  ? 

18.  What  may  be  the  dimensions  of  a  rectangle  whose 
area  is  equivalent  to  that  of  a  triangle  having  a  base  of 
6  in.  and  an  altitude  of  5  in.  ? 

19.  What  is  the  perimeter  of  a  square  equivalent  in  area 
to  a  triangle  whose  base  is  10  in.  and  altitude  5  in.  ? 

20.  What  is  the  ratio  of  a  square  6  in.  long  to  a  triangle 
twice  as  long  and  one-half  as  high  ? 

21.  What  is  the  ratio  of  a  rhomboid  whose  base  is  4  in. 
and  altitude  2  in.  to  a  triangle  having  an  equal  base  and 
an  equal  altitude  ? 

22.  What  may  be  the  dimensions  of  a  rhombus  equivalent 
in  area  to  a  triangle  whose  base  is  12  in.  and  altitude  6  in.  ? 

23.  What  is  the  altitude  of  a  triangle  12  in.  long  equiva- 
lent in  area  to  a  parallelogram  12  in.  long  and  8  in.  high  ? 

24.  What  equals  the  sum  of  the  areas  of  two  triangles, 
one  a  in.  by  h  in.,  and  the  other  c  in.  by  c?  in.  ? 

Arts.    —^  -f  ^-^  —  area  of  two  triangles. 


ADVANCED   ARITHMETIC.  171 

25.    What  equals  the  difference  in  area  of  two  triangles, 

the  smaller  one  being  m  ft.  by  n  ft.,  and  the  larger  x  ft. 

by  y  ft.  ? 

x'y      m'n 
Ans.    —^ ^7-  =  difference. 

Constructions.  —  1.  To  inscribe  a  square  in  a  circle.  See 
note,  p.  145. 

Draw  two  diameters  perpendicular  to  each  other.  Join 
their  extremities  by  chords. 


Practice  inscribing  squares. 

2.    To  inscribe  a  regular  hexagon  in  a  circle,  apply  the 
radius  as  a  chord  six  times  to  the  circumference. 
Practice  inscribing  hexagons. 

1.  Find  all  the  ratios  you  can.     See  p.  172. 

2.  a'  equals  what  part  oi  q? 

"  In  mathematics  I  had  many  truths  put  before  my  eyes,  and  con- 
cluded many  othefTs  from  them  by  analogy."  —  Descartes. 

"The  most  obtrusive  form  of  matter  is  the  solid,  and  for  this  reason 
it  is  that  form  which  is  first  cognized  by  the  infant  intellect  of  man- 
kind, and  thus  serves  as  tlie  basis  for  the  subsequent  recognition  of 
other  forms."  —  J.  B.  Stallo. 

"  Among  the  Greeks,  as  in  the  school  of  Pestalozzi,  .  .  .  mathe- 
matics were  drawn  back  to  tlie  primary  elements  of  education."  — 
Hamilton. 


172 


ADVANCED    ARITHMETIC. 


3.  What  is  the  ratio  of  o'  to  a'  ? 

4.  What  is  the  ratio  of  each  form  to  the  hexagon  r  ? 

5.  Find  equal  units. 

6.  I  equals  what  part  of  r  ?  i  equals  what  part  ?     What 
is  the  ratio  of  I  to  i? 


ADVANCED    ARITHMETIC. 


173 


Practice  constructing  octohedrons. 


174  ADVANCED   ARITHMETIC. 

Practice  constructing  icosahedrons. 


"  Imagination  expands,  diminishes,  molds,  and  refines,  as  the  case 
may  be,  materials  derived  from  the  world  of  fact  and  observation."  — 
John  Tyndall. 

"It  is  quite  a  mistake  to  consider  what  Tyndall  calls  imagination 
as  mere  fancy.  It  is  exactly  the  opposite  that  is  meant  —  full,  sen- 
suous contemplation."  —  Helmholtz. 

"The  kingdom  of  science,  then,  cometh  not  by  observation  and 
experiment  alone,  but  is  completed  by  fixing  the  roots  of  observation 
and  experiment  in  a  region  inaccessible  to  both,  and  in  dealing  with 
which  we  are  forced  to  fall  back  upon  the  picturing  power  of  the 
mind." — John  Tyndall. 

"The  greater  the  power  of  abstraction  and  generalization  which  is 
observed,  the  greater  is  the  power  of  representation  which  is  implied." 
—  John  Fiske. 


ADVANCED   ARITHMETIC. 


175 


Construct  a  hexagonal  prism. 

General  method  of  inscribing  any  regular  polygon  in  a 
circle  shown  by  inscribing  a  regular  pentagon. 


Use  the  dividers  and  mark  off  five  equal  divisions  on  a  line. 

Describe  a  circumference,  using  half  the  sum  of  the  five  divisions 
as  a  radius. 

From  A  and  B  as  centers,  and  with  AB  as  a  radius,  describe  arcs 
intersecting  at  C.  Draw  the  line  CD  through  the  second  division 
of  AB.     The  line  DA  is  the  side  of  the  pentagon. 

Any  regular  polygon  can  be  inscribed  by  the  method  given 
above  by  making  the  number  of  divisions  of  the  diameter  equal 
to  the  number  of  the  sides  of  the  polygon  and  drawing  CD 
through  the  second  division  of  the  diameter. 


-X 


176 


ADVANCED    ARITHMETIC. 


This  diagram  represents  one-half  of  the  surface  of  a  dodecahedron. 


Construct  dodecahedrons.    Construct  cones  and  cylinders. 


ADVANCED    ARITHMETIC. 


177 


1.  Practice  drawing  the  following  (see  method,  p.  175)  : 
A  regular  hexagon  ;  a  regular  heptagon  ;  a  regular  octa- 
gon, etc. 

2.  In  and  out  of  school  find  regular  polygons.  Observe 
the  drawing  models. 

3.  Draw  pentagon  and  five-pointed  star. 

4.  Pupils  give  directions  for  inscribing  figures  and  other 
pupils  draw  figures.  Inscribe  the  figures  mentally  accord- 
ing to  oral  directions,  and  tell  what  polygon  is  inscribed. 


Regular  Polygons.  —  1.    In  what  are  the  above   regular 
polygons  alike  ? 

2.  What  five  things  in  each  of  the  regular  polygons  are 
respectively  equal  ? 

3.  ^  is  a  greater  radius  of  a  regular  polygon.  Show 
greater  radii  in  the  other  polygons.  What  is  a  greater 
radius  ? 

4.  Into  what  do  the  greater  radii  of  a  regular  polygon 
separate  the  polygon  ? 

5.  B  is  a  less  radius.  Find  less  radii  of  other  poly- 
gons.    What  is  a  less  radius  ? 

6.  In  which  regular  polygon  is  there  the  greatest 
difference  in  the  lengths  of   the  greater  and   less  radii? 

7.  As  the'' number  of  sides  of  the  polygons  increase, 
what  change  takes  place  in  the  relative  lengths  of  the 
greater  and  less  radii  ? 

8.  Which  line  of  the  regular  polygon  is  the  altitude  of 
the  triangles  into  which  it  may  be  separated  ? 

9.  Can  you  construct  a  polygon  with  equal  greater  and 
equal  less  radii  and  not  make  it  regular  ? 


178 


ADVANCED   ARITHMETIC. 


10.  If  the  greater  and  less  radii  of  any  polygon  are 
respectively  equal,  what  kind  of  a  polygon  is  it  ? 

11.  What  is  a  regular  polygon  ? 

A  regular  polygon  is  a  polygon  in  which  the  greater  and 
less  radii  are  respectively  equal. 

12.  In  which  regular  polygon  are  all  the  radii  equal  ? 
A  circle  is  a  regular  polygon  whose  radii  are  equal. 

13.  How  may  the  area  of  any  regular  polygon  be  found  ? 


m 


B 


Regular    Polygons   and   Equivalent   Rectangles.  —  1.    Tri- 
angle A  is  separated  into  how  many  equal  triangles  ? 

2.  Show  the  circumference  of  A.  Which  line  of  rect- 
angle B  equals  the  circumference  of  ^  ?  The  altitude  of 
B  equals  one-half  of  what  line  in  A  ?  What  is  true  of  A 
and  B  ? 

3.  Draw  a  regular  pentagon.  Draw  a  line  equal  to  the 
circumference  of  the  pentagon.  On  the  line,  construct  a 
rectangle  equivalent  to  the  pentagon. 

4.  Draw  a  hexagon.  Draw  a  line  equal  to  the  circum- 
ference of  the  hexagon.  On  the  line,  construct  a  rectangle 
equivalent  to  the  hexagon. 


ADVANCED   AKTTHMETiC.  179 

5.  Draw  a  regular  heptagon.  Draw  a  less  radius  of  the 
heptagon,  but  do  not  separate  the  heptagon  into  triangles. 
Draw  a  line  equal  to  the  circumference  of  the  heptagon. 
Construct  on  the  line  a  rectangle  equivalent  to  the 
heptagon. 

6.  If  the  number  of  units  in  the  circumference  of  a 
regular  polygon  is  given,  what  other  number  is  required  in 
order  to  find  the  area  of  an  equivalent  rectangle  ? 

The  number  of  units  in  the  area  of  the  polygon  equals 
the  product  of  what  numbers  ?     Why  ? 

7.  Is  a  circle  a  regular  polygon  ?  When  the  number 
of  units  in  the  circumference  of  the  circle  is  known,  what 
other  number  is  needed  to  find  its  area  ? 

The  product  of  what  numbers  equals  the  number  of  units 
in  the  area  of  the  circle  ? 

8.  If  a  is  the  perimeter  of  an  equilateral  triangle  and 
h  its  less  radius,  what  equals  its  area  ? 

9.  If  a  is  the  perimeter  of  a  square  and  h  its  less  radius, 
what  equals  its  area  ? 

10.  If  a  is  the  perimeter  of  a  regular  hexagon  and  b  its 
less  radius,  what  equals  its  area  ? 

11.  If  <x  is  the  perimeter  of  a  regular  octagon  and  h  its 
less  radius,  what  equals  its  area  ? 

12.  If  a  is  the  circumference  of  a  circle  and  b  its  radius, 
what  equals  its  area  ? 

13.  If  the  length  of  one  side  of  an  equilateral  triangle  is 
a  and  its  less  radius  b,  what  equals  its  area  ? 

14.  If  the  li^ngth  of  one  side  of  a  square  is  a,  what  equals 
its  area  ? 

15.  What  is  the  ratio  of  the  perimeter  of  a  square  to  its 
less  radius  ? 

Areas  of  Regular  Polygons.  —  1.    The  circumference  of  a 
square  equals  how  many  times  its  less  radius  ? 


180 


ADVANCED    ARITHMETIC. 


2.  If  5  is  the  less  radius  of  a  square, 
what  is  its  circumference  ? 

The  less  radius  equals  what  part  of  the 
circumference  of  a  square  ? 

3.  If  56  is  the  circumference  of  a  square, 
what  is  its  less  radius  ?  if  19  ?  it  x? 

4.  What  is  the  ratio  of  the  less  radius 
of  a  square  to  its  circumference  ? 

5.  What  is  the  radius  of  the  largest  cir- 
cle that  can  be  inscribed  in  a  square  8"  in 
diameter  ? 

6.  What  is  the  less  radius  of  a  square 
5  ft.  long  ?  of  a  square  1  mile  in  circum- 
ference ?  1  yd.  in  circumference?  f  in.  in 
circumference?  .35  ft.  in  circumference? 

7.  What  is  the  less  radius  of  a  square 
whose  circumference  is  100  ? 

8.  What  is  the  less  radius  of  a  square 
whose  area  is  16  ?  25  ?  49  ? 

9.  Draw  a  rectangle  which  is  equivalent 
to  a  square  and  which  has  a  base  equal  to 
the  circumference  of  the  square.  What  is 
its  altitude  ?  What  is  the  area  of  this 
rectangle  ? 

10.  In  the  manner  of  the  last  problem 
find  the  area  of  a  6"  square  ;  a  12"  square  ; 
a  3  ft.  square ;  a  1"  square ;  a  9"  square ; 
a  square  5§"  long. 

11.  The  perpendicular  distance  from  the 
center  of  a  square  floor  to  one  of  its  sides 
is  12  ft.  What  is  the  area  of  the  floor  ? 
What  is  the  length  of  the  perimeter  of  the 
floor  ?  If  the  room  is  13  ft.  high,  what  is 
the  area  of  the  walls  of  the  room  ? 


ADVANCED    ARITHMETIC.  181 

1.  Cut^a  circle    having  a  7"  diameter.     Measure   the 
circumference  carefully. 

Suggestion.  —  Bend  a  strip  of  paper  around  half  of  the  circum- 
ference, then  measure  the  paper.  Twice  the  length  of  the  paper 
equals  the  circumference. 

2.  What  is  the  length  of  the  circumference  of  a  7" 
circle  ?  The  circumference  of  a  circle  is  how  many  times 
as  long  as  \  of  its  diameter  ?  The  diameter  equals  how 
many  twenty-seconds  of  its 
circumference  ? 

3.  How  many  equal  divi- 
sions in  the  diameter  of  the 
circle  m  ? 

How  many  divisions  equal 
to  1  of  the  diameter  are  found 
in  -J-  of  the  circumference  ? 
in  the  circumference  ? 

4.  What  is  the  ratio  of 
the  circumference  of  a  circle 
to  its  diameter  ?     What  is  ^ 

the  ratio  of  the  diameter  to  the  circumference  ? 

5.  If  the  diameter  of  a  circle  is  known,  what  can  be 
found  ?     How  ? 

6.  If  the  circumference  of  a  circle  is  given,  what  can 

be  found  ?     How  ? 

♦ 

7.  Draw  a  straight  line  equal  to  the  circumference  of  a 
circle  S^"  in  diameter. 

8.  Draw  a  straight  line  equal  to  the  circumference  of 
the  circle  If"  in  diameter. 

9.  The   circumference   of   a   circle   equals  how  many 
diameters  ? 

What  is  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter  ? 


182  ADVANCED   ARITHMETIC. 

Note.  —  3f  is  an  approximate  value  of  the  ratio  of  the  circum- 
ference of  a  circle  to  its  diameter.  Its  absolute  value  cannot  be 
expressed.  For  exact  computations  this  value  has  been  carried  out 
many  decimal  places.  ^\  is  accurate  enough  for  most  purposes. 
For  more  precise  estimates  the  decimal  3.1416  may  be  used. 
This  ratio  may  be  denoted  by  the  Greek  letter  tt  (pi);  thus,  the 
circumference  of  the  circle  may  be  indicated  by  tt  cZ  or  27r  r  (d 
diameter,  and  r  radius). 

10.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is  14"?  21"?  9"?  42"?  6J"?  Hi"?  I"?  84"?   1.38"? 

22 '  14 

Ux. :  —  — - —  =  number  of  in.  in  the  circumference  of  a 

circle  whose  diameter  is  14". 

11.  What  is  the  circumference  of  a  circle  whose  radius 

is  14"  ?   6"?   13"?   21"?    7"?    y"?    7"?   iy'9 

22 • 2  •  14 
Ex. :  — =  the  number  of  in.  in  the  circumference 

of  a  circle  whose  radius  is  14". 

Draw  circles  of  different  sizes  on  the  blackboard.  Measure 
diameters.     Pupils  in  seats  find  circumferences. 

12.  If  the  hour-hand  of  a  clock  is  5"  long,  how  far  does 
its  extremity  move  in  12  hr.  ?  in  8  hr.  ?  in  Ij  hr.  ? 

13.  A  horse  is  tied  to  a  stake  by  a  strap  8  rd.  long. 
What  is  the  circumference  of  the  circle  in  which  he  has 
grazed  ? 

14.  The  diameter  of  a  circle  is  18  rd.  How  far  will  John 
have  to  travel  to  pass  around  the  circle  if  he  starts  at  the 
center  ? 

15.  What  is  the  circumference  of  a  cylinder  if  its  diam- 
eter is  7"?    15"?    I"? 

16.  What  is  the  diameter  of  a  circle  whose  circumfer- 
ence is  3"  ?  3J-"?  7"?  8"?  1ft.?  1"?  98"?  49"?  22ft.? 
3  ft.? 


ADVANCED    ARITHMETIC.  183 

17.  The  diameter  of  a  circle  is  7";  what  is  the  length  of 
an  arc  of  75°?  of  170°? 

18.  What  is  the  length  of  an  arc  of  75°  in  a  circle  whose 
diameter  is  16  ft.  ? 

19.  What  is  the  diameter  of  that  circle  the  length  of 
whose  arc  of  80°  is  44  in.  ? 

20.  How  many  degrees  in  the  arc  whose  length  is  66 
in.  if  the  diameter  of  the  circle  is  60  in.  ? 

21.  What  is  the  diameter  of  a  tree  whose  circumference 
is  20"  ?  What  is  the  diameter  of  a  log  whose  circumfer- 
ence is  6  ft.  ? 

12.    Draw  a  circle  whose  circumference  is  5^  ft. 
Draw  a  straight  line  equal  to  the  circumference,  then  a 
line  equal  to  the  diameter. 

How  long  is  the  radius  ?     Describe  the  circle. 

23.  Draw  a  line  equal  to  the  circumference  of  a  6" 
circle. 

24.  A  line  2f  ft.  long  equals  the  circumference  of  a 
circle  whose  diameter  is  how  many  inches  ? 

25.  Draw  a  circle  having  a  circumference  of  11  in. 

26.  If  7  in.  is  the  greatest  distance  through  a  ball,  what 
is  the  greatest  distance  around  the  ball  ? 

27.  If  3  in.  is  the  diameter  of  a  croquet  ball,  what  is  its 
circumference  ? 

28.  If  25  is  the  number  of  units  in  the  area  of  a  square, 
what  is  the  square  root,  or  the  number  of  units  in  its 
length  ? 

29.  What  is-  the  square  root  of  17  ?  {Ans.  4  +.)  of 
.17?   {Ans.    .4+.) 

30.  What  is  the  square  root  of  i?  of  i?  of  |?  of  ^^  ? 

31.  What  is  the  square  root  of  16  ?  of  36  ?  of  64  ?  of 
49  ?   of  81  ?  of  84  ?  of  .84  ? 

32.  Vl44  =  12,  Vi69  =  ?  V72  =  ?  Vl3  =  ?  V95  =  ? 
Vi47  =  ?   ViOO  =  ? 


184 


ADVANCED    ARITHMETIC. 


1.  Draw  the  square  of  the  d  of  the  circle  whose  d  is 
1";  2";  2i". 

2.  A  square  1"  long  equals  what  part  of  a  square  2" 
long  ?  of  a  square  3"  long  ?  A  square  2"  long  equals  what 
part  of  a  square  3"  long  ? 

3.  A  square  -J-"  long  equals  what  part  of  a  square  1" 
long  ? 

4.  A  square  f "  long  equals  what  part  of  a  square  1" 
long  ?  2"  long  ? 

5.  The  square  of  the  radius  of  a  circle  equals  what 
part  of  the  square  of  the  diameter  ? 

6.  A  rectangle  3|"  long  by  :J-"  wide  equals  what  part 
of  a  square  inch  ? 

7.  If  ^  is  the  diameter  of  a  circle,  what  is  its  circum- 
ference ? 

8.  If  b  is  the  circumference  of  a  circle,  what  is  its 
diameter  ? 


i 

fe 

^ 

^ 

^ 

11 

11 

14 

\ 

14 

1- 

f 

^^^M 

^^^M 

^^^W 

1 

+d 


9.  Draw  a  circle.  Draw  a  straight  line  equal  to  the 
circumference  of  the  circle.  On  the  straight  line  construct 
a  rectangle  equivalent  to  the  circle. 

10.  Draw  the  square  of  the  diameter  of  the  circle. 

11.  The  rectangle  equivalent  to  the  area  of  the  circle 
equals  what  part  of  the  square  of  its  diameter  ?     What  is 


ADVANCED    ARITHMETIC. 


185 


the  ratio  of  the  area  of  the  circle  to  the  square  of  its 
diameter  ?  Of  any  circle  to  the  square  of  its  diameter  ? 
What  is  the  ratio  of  c?^  to  the  area  of  the  circle  ? 

12.  What  is  the  area  of  a  circle  whose  diameter  is  7"? 
What  is  the  square  of  the  diameter  ?  The  circle  equals 
what  part  of  the  square  of  its  diam- 
eter ?  38-^  sq.  in.  equals  what  part 
of  49  sq.  in.  ? 

13.  The  circle  B  is  what  part  of 
the  square  of  its  diameter?  The 
sum  of  the  parts  of  the  square  out- 
side of  the  circle  equals  what  part 
of  the  square  ?  what  part  of  the 
circle  ? 

14.  If  77  is  the  number  of  units  in  the  area  of  the 
circle  B,  what  equals  the  number  in  the  area  of  the 
square  ? 

15.  If  21  is  the  number  of  units  in  the  diameter  of  a 
circle,  what  equals  the  number  of  units  in  its  area  ? 
11 '21  21 


14 


=  the  number  of  units  in  its  area. 


1-1  of  212  =  what  ? 


21  ■  21  =  the  square  of  what  ? 

16.    What  equals  the  number  of  units  in  the  areas  of 


''//////m!?;>M7////^//L^///^M 


circles  having  the  following  diameters  :   8",  14",  24",  85", 
.7",  2.1",  3^",  /t-",  1.38"? 


186 


ADVANCED    ARITHMETIC. 


1/ 

/      / 

17.  Find  all  the  ratios  you  can.  What  is  the  ratio  of  d 
to  k?  The  circle  inscribed  in  d  equals  what  part  of  ti  ? 
The  circle  inscribed  in  k  equals  what  part  of  k  ?  What, 
then,  is  the  ratio  of  the  larger  circle  to  the  smaller  ?    Why  ? 

18.  What  is  the  ratio  of  circles 
having  the  following  diameters : 
1"  and  2"?  2"  and  4"?  2"  and 
3'?  2'  and  5'?  f  and  |"?  ^3^' 
and  I'? 

19.  What  is  the  relation  of  the 
circle  to  the  inscribed  square  and 
to  the  circumscribed  square  ?  Of 
the    circumscribed    square    to    the 

circle  ?  to  the  inscribed  square  ?     Of  the  inscribed  square 
to  the  circle  ?  to  the  circumscribed  square  ? 

20.  Draw  circles  on  the  blackboard.  Measure  their 
diameters  and  compute  areas. 

21.  Make  many  measures  of  the  diameters  of  cylinders, 
cups,  barrels,  pails,  bases  of  cones,  plane  surfaces  of  hemi- 
spheres, dials  of  clocks  and  watches,  etc.  Compute  the 
areas  of  the  circles  whose  diameters  are  measured. 

22.  A  sector  is  a  part  of  a  circle  bounded  by  two  radii 
and  an  arc.     Show  me  a  sector. 

A  boy  receives  a  sector  of  a  pie  having  an  arc  of  100 
degrees.     What  part  of  the  pie  does  he  receive  ? 

23.  If  the  minute  hand  of  a  clock  is  5J"  long,  what  is 
the  area  of  the  circle  that  it  describes  in  1  hr.  ? 

24.  Draw  a  circle  having  a  diameter  of  2.76".  What  is 
the  area  of  the  circle  ? 

25.  From  a  rectangle  2"  by  3",  how  many  circles  1"  in 
diameter  can  be  cut  ?  After  cutting  the  circles,  what  part 
of  the  rectangle  remains  ? 

2(j.    If  a  is  the  diameter  of  a  circle,  what  equals  its  area  ? 


ADVANCED    AKITHMETTC.  187 

27.  The  square  of  the  diameter  of  a  circle  equals  what 
part  of  the  square  of  its  circumference  ? 

28.  The  diameter  of  the  base  of  a  cylinder  is  3f " ;  its 
altitude  is  7"  ;  what  is  its  entire  surface  ? 

29.  If  a  is  the  radius  of  a  circle,  what  is  its  area  ? 

30.  If  a  is  the  area  of  a  circle,  what  equals  the  diame- 
ter of  the  circle  ? 

31.  If  a  is  the  area  of  a  circle,  what  equals  the  radius 
of  the  circle  ? 

32.  If  a  is  the  square  of  the  radius  of  a  circle,  what 
equals  the  area  of  the  circle  ? 

33.  If  a  is  the  square  of  the  diameter  of  a  circle,  what 
equals  the  area  of  the  circle  ? 

34.  If  a  is  the  perimeter  of  the  square  of  the  diameter 
of  a  circle,  what  equals  the  area  of  the  circle  ? 

35.  To  what  rectangle  is  a  circle  equivalent  ? 

36.  To  what  rectangle  is  any  regular  polygon  equiva- 
lent? 

37.  What  are  the  dimensions  of  a  rectangle  equal  in  area 
to  a  circle  whose  diameter  is  1  in.  ? 

38.  What  may  be  the  dimensions  of  a  rectangle  equal  in 
area  to  a  circle  whose  radius  is  ^  in.  ? 

39.  What  may  be  the  dimensions  of  a  rectangle  equal  in 
area  to  a  circle  whose  diameter  is  2  in.  ? 

40.  If  a  is  the  square  of  the  radius  of  a  circle,  what  is 
the  square  of  its  diameter  ? 

41.  If  «-  is  the  square  of  the  diameter  of  a  circle,  what 
equals  the  sqiaare  of  its  radius  ? 

42.  What  is  the  area  of  the  largest  circle  that  can  be 
inscribed  in  a  2-in.  square? 

43.  What  is   the   area   of   the   largest  circle  that   can 
be  inscribed  in  a  room  x  ft.  long  and  7  ft.  wide  ? 

44.  What  is  the  circumference  of  a  circle  that  can  be 


188  ADVANCED   ARITHMETIC. 

traced  by  a  pair  of  compasses  whose  points  are  x  inches 
apart  ? 

45.  What  equals  the  area  of  the  circle  in  question  44  ? 

46.  What  is  the  area  of  the  largest  circle  that  can  be 
inscribed  in  a  3-in.  square  ? 

47.  What  is  the  ratio  of  a  circle  inscribed  in  a  1-in, 
square  to  a  circle  inscribed  in  a  2-inch  square  ? 

48.  What  is  the  ratio  of  a  circle  inscribed  in  a  2-in. 
square  to  a  circle  inscribed  in  a  4-inch  square  ? 

49.  What  is  the  area  of  a  circle  whose  circumference 
is  88"? 

50.  If  this  room  is  30  ft.  long  and  25  ft.  wide,  what  is 
the  area  of  the  largest  circle  that  can  be  described  in  the 
room  ? 

51.  What  is  tlie  area  of  a  railroad  turn-table  35  ft.  in 
diameter  ? 

52.  The  square  of  the  radius  of  a  circle  equals  what  part 
of  the  square  of  the  diameter  ? 

53.  A  log  is  15  ft.  long  and  the  diameter  of  each  end 
is  4f  ft.     What  is  its  entire  surface  ? 

54.  What  is  the  relative  size  of  two  circles,  one  of  whicli 
has  a  diameter  of  f "  and  the  other  a  diameter  of  |"  ? 

55.  If  X  is  the  radius  of  a  circle,  find  area. 

56.  If  X  is  the  diameter,  find  radius.  Find  circum- 
ference.    Find  area. 

57.  If  X  is  the  circumference  of  a  circle,  find  diameter, 
radius,  area. 

58.  If  a  is  the  diameter  of  the  base  of  the  cylinder  and 
h  is  its  altitude,  find  its  lateral  surface ;  its  entire  surface. 

59.  If  a  is  the  diameter  of  the  base  of  a  cone  and  h  the 
slant  height,  find  lateral  surface  ;  entire  surface. 

60.  What  is  the  entire  surface  of  a  cylinder  whose  base 
is  13"  in  diameter  and  whose  altitude  is  17"?  Find  tlie 
lateral  surface. 


ADVANCED    ARITHMETIC.  189 

61.  What  is  the  entire  surface  of  a  cone  whose  base  is 
13"  in  diameter  and  whose  slant  height  is  17"  ? 

62.  What  is  the  entire  surface  of  a  cylinder  if  the 
diameter  of  the  base  is  1  ft.  and  the  altitude  1  ft.? 

63.  A  circle  whose  diameter  is  1"  equals  what  part  of  a 
circle  whose  diameter  is  1^"? 

64.  Grive  two  methods  of  finding  area  of  a  circle. 

Volumes  of  Cylinders.  —  1.  The  largest  cylinder  that  can 
be  cut  or  turned  out  of  a  cubic  inch  equals  what  part  of  the 
cubic  inch  ?  ^^__ 

2.  The  largest  cylinder  that  can  |r  '  ^^^^^ 
be  turned  out  of  any  square  prism  |  JH 
equals  what  part  of  it  ?                           I           ^^|pl|l|^ 

3.  What  equals  the  number  of 

cubic  inches  in  the  largest  cylinder  that  can  be  turned  out 
of  a  cubic  foot  ? 

4.  Give  a  rule  for  finding  the  volume  of  a  cylinder. 

5.  Keview  from  the  first  of  mensuration. 

6.  What  is  the  volume  of  a  cylinder  whose  altitude  is 
5"  and  the  diameter  of  whose  base  is  2.76"  ? 

7.  What  is  the  volume  of  a  square  prism  7"  long  and 
14"  high?    . 

8.  What  is  the  volume  of  a  cylinder  whose  altitude  is 
14"  and  the  diameter  of  whose  base  is  7"  ? 

9.  If  h  is  the  altitude  of  a  cylinder  and  a  is  the  diameter 
of  its  base,  what  equals  its  volume  ? 

10.  If  51"  i»  the  diameter  of  a  cylinder  and  its  altitude 
is  3.7",  what  is  its  volume  ? 

11.  The  volume  of  the  cylinder  in  problem  8  equals  what 
part  of  the  prism  in  problem  7  ? 

12.  How  many  2"  cubes  can  be  put  into  a  box  5"  X  7"  X  9"  ? 
What  is  the  volume  of  the  largest  cylinder  that  can  be  put 
into  the  box  ? 


190 


ADVANCED    ARITHMETIC. 


ADVANCED   ARITHMETIC.  191 

Polyhedrons.  —  What  is  a  geometrical  solid  ?  In  what 
are  these  solids  alike  ? 

What  is  a  polygon  ?  By  what  are  polyhedrons  bounded  ? 
What  is  a  polyhedron  ? 

A  polyhedron  is  a  solid  bounded  by  polygons. 


PKISMS. 


Prisms. — 1.  In  what  are  these  polyhedrons  alike  ?  differ- 
ent? 

2.  What  is  true  of  the  upper  and  lower  bases  of  these 
polyhedrons  ? 

3.  Of  what  is  the  convex  surface  of  each  composed  ? 

4.  Show  the  equal  and  parallel  polygons.  Show  the 
parallelograms. 

5.  These  polyhedrons  are  called  prisms.  What  is  a 
prism  ?  A  prism  is  a  polyhedron  whose  bases  are  equal 
and  parallel  and  whose  convex  surface  is  composed  of 
parallelograms. 

6.  What  is  the  least  number  of  lateral  surfaces  a  prism 
can  have  ?     The  greatest  number  ? 

7.  What  is  the  name  of  the  prism  having  the  least  num- 
ber of  sides  ?     The  greatest  number  ? 

8.  Is  a  cubg  a  prism  ?     Why  ?     Is  a  cylinder  ?     Why? 

Volumes  of  Prisms.  — Make  of  <^  tough  check,"  or  of  some 
other  tough  card  board,  a  quadrangular,  a  triangular,  and 
a  cylindrical  prism.  Make  their  bases  equivalent  and  alti- 
tudes equal.  The  diagrams  (see  page  192)  will  aid  in 
making  the  prisms.  Do  the  work  carefully.  Compare 
their  volumes  by  measuring  sand,  salt,  or  sugar. 


192 


ADVANCED    ARITHMETIC. 


PRISMS. 


1.  Eind  the  areas  of  the  bases  of  the  prisms. 

2.  What  is  true  of  the  bases  of  the  prisms  ?  of  their 
altitudes  ?     of  their  volumes  ? 


ADVANCED   ARITHMETIC.  193 

If  the  quadrangular  prism  were  1"  high,  how  many 
cubic  inches  would  it  contain  ? 

If  the  triangular  prism  were  1"  high,  how  many  cubic 
inches  would  it  contain  ? 

If  the  cgllnder  were  1"  high,  how  many  cubic  inches 
would  it  contain  ? 

Each  prism  is  5"  high.  How  many  cubic  inches  does 
each  contain  ? 

What  is  the  number  of  sq.  in.  in  the  base  of  each 
prism  ?  What  is  the  number  of  inches  in  the  altitude  of 
each  ?  Then  the  product  of  what  numbers  equals  the  num- 
ber of  units  in  the  volume  of  each  ? 

3.  What  is  a  prism  ?     When  are  prisms  equivalent  ? 

4.  What  shows  the  relation  of  prisms  having  equiva- 
lent bases  ?    having  equal  altitudes  ? 

5.  If  a  and  h  are  two  prisms  having  equal  altitudes  and 
the  base  of  a  equals  |  of  the  base  of  b,  what  is  the  relation 
oihto  a? 

Suggestion Draw  the  diagrams  on  pages  192  and  194,  take 

them  to  a  tinsmith,  and  have  the  soHds  made.  The  making  of 
the  three  prisms  and  of  the  two  pyramids  will  cost  less  than  75^. 
The  comparison  of  the  volume  by  measuring  water  will  interest 
the  class  and  explain  much  of  the  work  that  follows. 

Pyramids.  —  1.    In  what  are  these  polyhedrons  alike  ? 
2.   What  general  name  do  we  give  to  the  base  of  each  ? 


3.  Of  what  is  the  lateral  surface  of  each  composed? 

4.  These  polyhedrons  are  pyramids. 


194 


ADVANCED   AHITHMETIC. 


5.  The  base  of  which  pyramid  has  the  least  number  of 
sides  ?     The  greatest  number  ? 

6.  If  the  base  of  a  cone  is  a  polygon,  of  what  is  the 
lateral  surface  of  the  cone  composed  ? 

7.  What  is  a  pyramid  ? 

A  pyramid  is  a  polyhedron  having  a  polygon  for  its  base 
and  triangles  for  its  lateral  surface. 

8.  What  is  the  altitude  or  height  of  a  pyramid  ? 

Out  of  "  tough  check  "  make  a  triangular  pyramid  and  a  conical 
pyramid  having  equivalent  bases  and  equal  altitudes.  Compare 
the  volumes  of  the  two  pyramids. 

The  diagrams  below  show  how  the  tough  check  may  be  cut  and 
folded  into  the  required  pyramids. 

To  make  a  pyramid  and  a  cone  having  equivalent  bases 
and  equal  altitudes. 


Draw  a  line  hd. 

Draw  ac  perpendicular  to  hd  at  c. 

Draw  ah  and  ad. 

With  a  as  a  center  and  ah  as  a  radius  draw  the  arc  ef. 
With  (^  as  a  center  and  a  5"  radius  draw  the  arc  07i  inter- 
secting ef. 

Join  a  and  e  and  de. 
Cut  and  fold  to  shape. 

What  is  the  altitude  of  the  cone  ?  of  the  triangular  pyr- 
amid ?     What  is  the  base  of  the  cone  ? 


ADVANCED   ARITHMETIC. 


195 


Suggestion.  —  It  may  be  well  to  make  the  slant  height  of  the 
cone  5.2''  and  the  circumference  of  the  base  8.7''  instead  of  the 
more  accurate  measures  given  above. 

1.    What  is  true  of  the  bases  of  the  triangular  and  con- 
ical pyramids  ?    of  their  altitudes  ?   of  their  volumes  ? 


PYEAMIDS. 

2.  Compare  the  volumes  of  the  pyramids  and  prisms 
by  measuring  water. 

3.  What  is  the  ratio  of  one  of  the  prisms  to  one  of  the 
pyramids  ? 

4.  How  is  the  volume  of  a  prism  found  ?     How,  then, 
may  the  volume  of  the  pyramid  be  found  ? 


196 


ADVANCED    ARITHMETIC. 


ADVANCED    ARITHMETIC. 


197 


5.  What  is  the  volume  of  a  pyramid  whose  base  is  6 
sq.  in.  and  altitude  5"  ? 

6.  The  base  of  a  pyramid  is  75  sq.  in.  and  its  altitude 
27".     What  is  its  volume  ? 

7."  If  the  base  of  a  cone  is  18  sq.  in.  and  its  altitude 
9",  what  is  its  volume  ?  What  is  the  ratio  of  the  volume 
of  a  prism  whose  base  is  18  sq.  in.  and  altitude  3"  to  the 
cone  ? 


8.    If  the  diameter  of  the  base  of  a  cone  is  7"  and  its 
altitude  is  8",  what  is  its  volume  ? 

R  '  1 1  •  ^  *  7 

the  number  of  units  in  the  volume  of  the  cone. 


8 ' 11 " 7  '  7 
what  ?     Then  —. equals  what  ?     Then 


3- 

L4 

11 

.•7 

'  7 

14 

8 

11 

•7 

7 

14 


14 


what? 


198 


ADVANCED    ARITHMETIC. 


9.    Find  the  volumes  of  the  following  cones  : 

Diameter  of  Base.  Altitude, 

21"  12" 

63"  18" 

15"  17" 

W  5" 

.9"  1.1" 

.18"  .84" 

10.  Make  and  solve  five  problems  similar  to  those  given 
above.  Measure  the  diameters  of  the  bases  and  the  alti- 
tudes of  cones  and  find  volume. 

11.  Draw  a  quadrangular  prism  equal  to  the  sum  of  the 
following  pyramids  :  The  altitude  of  each  is  6" ;  the  bases 
are  respectively  7  sq.  in.,  5  sq.  in.,  and  3  sq.  in. 

12.  Draw  a  quadrangular  prism  equal  to  a  pyramid  whose 
base  is  6  sq.  in.  and  whose  altitude  is  1^". 

13.  Give  a  method  of  finding  the  volume  of  three  pyra- 
mids of  equal  altitude.  Would  this  do?  Multiply  the 
number  of  units  in  the  sum  of  their  bases  by  ^  of  the 
number  of  units  in  their  altitude. 

14.  Given  100  pyramids  of  an  equal  altitude,  what 
equals  the  sum  of  their  volumes  ? 

15.  How  may  the  relative  magnitude  of  prisms  which 
have  equal  altitudes  be  found  ? 

16.  If  a  and  h  are  two  prisms  having  equal  altitudes  and 


PRISMS  AND  PYKAMIDS. 


ADVANCED    ARITHMETIC.  199 

the  base  of  a  equals  |  of  the  base  of  h,  what  is  the  relation 
of  a  to  ^^  ? 

17.  What  is  a  prism  ?     When  are  prisms  equivalent  ? 

18.  How  may  the  volume  of  a  prism  be  found  ?  of  a 
pyramid  ? 

19.  Draw  a  rectangular  prism  equivalent  to  a  pyramid 
having  a  base  of  6  sq.  in.  and  an  altitude  of  4^". 

20.  What  are  the  dimensions  of  a  rectangular  solid 
equivalent  to  a  triangular  pyramid  having  a  base  of  12 
sq.  in.  and  an  altitude  of  15"  ? 

21.  What  are  the  dimensions  of  a  rectangular  solid  equiv- 
alent to  a  cone  whose  base  equals  12  sq.  in.  and  altitude  15"  ? 

22.  The  largest  cone  that  can  be  turned  out  of  a  cubic 
foot  of  wood  equals  what  part  of  the  cubic  foot  ? 

23.  How  many  cubic  inches  in  the  largest  cone  that  can 
be  turned  out  of  a  cubic  foot  of  wood  ? 

24.  The  largest  pyramid  that  can  be  cut  out  of  a  square 
prism  equals  what  part  of  the  prism  ? 

25.  What  is  the  volume  of  a  pyramid  whose  base  is  84 
sq.  in.  and  whose  altitude  is  72"  ? 

26.  What  is  the  volume  of  a  cone  if  its  altitude  is  35  ft. 
and  the  diameter  of  its  base  is  15  ft.  ? 

27.  What  equals  the  volume  of  a  conical  pyramid  if  its 
altitude  is  a  and  the  diameter  of  its  base  is  6  ? 

«ll-^^_o 

3  14         • 

28.  What  is  the  volume  of  a  cone  if  the  diameter  of  its 
base  is  7"  and  its  altitude  9"? 

29.  What  is  the  volume  of  a  cone  if  the  diameter  of  its 
base  is  29"  and  its  altitude  28"  ? 

30.  If  the  circumference  of  the  base  of  a  cone  is  54"  and 
-its  altitude  is  12",  what  is  its  volume? 

'      31.    How  many  yards  of  canvas  f  yd.  wide  are  required  to 
make  a  conical  tent,  7  ft,  slant  height,  and  8  ft.  in  diameter  ? 


200 


ADVANCED   ARITHMETIC. 


^ 


ADVANCED    ARITHMETIC. 


201 


32.  What  equals  the  volume  of  x  pyramids  of  equal 
altitude,  supposing  the  sum  of  their  bases  to  be  30  sq.  in. 
and  the  altitude  8"  ? 

33.  How  can  the  area  of  the  base  of  a  pyramid  be  found 
if  the  number  of  units  in  its  volume  and  the  number  of 
units  in  its  altitude  are  given  ?  How  can  the  volume  of  a 
pyramid  be  found  if  the  number  of  units  in  its  altitude 
and  the  number  of  units  in  its  base  are  given  ? 

Regular  Polyhedrons.  —  1.  In  what  are  the  polyhedrons 
on  page  200  alike  ? 

2.  Describe  the  greater  radius  of  each.     What  is  true 
of  the  greater  radii  of  each  ? 

3.  Describe  the  less  radius  of  each.     What  is  true  of 
the  less  radii  of  each  respectively  ? 


202 


ADVANCED    ARITHMETIC. 


4.  In  the  regular  polyhedron  M,  aissi  less  radius  and  b 
a  greater  radius.  Show  the  greater  and  less  radii  of  this  room. 

5.  Are  all  of  the  greater  radii  of  the  room  equal  ?    Are 
all   of  the  less  radii  equal?     Look  carefully.     Is  it  the 


KEGULAK  POLYHEDRONS. 


same  distance  from  the  center  of  the  room  to  each  of  the 
four  walls  as  to  the  ceiling  and  to  the  floor  ? 

6.  What  is  true  of  the  greater  and  less  radii  respec- 
tively of  a  regular  polyhedron  ? 

7.  What  is  a  regular  polyhedron  ? 

A  regular  polyhedron  is  a  solid  whose  greater  and  less 
radii  are  respectively  equal. 

8.  In  which  solid  is  there  the  greatest  difference  in 
the  length  of  the  greater  and  less  radii  ?  In  which  are  all 
the  radii  equal  ? 

9.  What  things  are  respectively  equal  in  a  regular 
polygon?     In  a  regular  polyhedron? 

10.    Into  what  equal  solids  can  a  regular  polyhedron  be 
separated  ? 


ADVANCED    ARITHMETIC.  203 

1.  Into   how   many   equal    pyramids   can   a   cube   be 
separated  ? 

2.  In  this  cube,  find  the  edges  and  the  base  of  each  of 
the  six  pyramids  into  which  the  cube  may  be  separated. 


^A? 


3.  Find  the  edges  and  the  base  of  each  of  the  six 
pyramids  into  which  this  room  may  be  separated. 

4.  What  is  the  name  of  the  altitude  of  the  pyramids 
into  which  a  regular  polyhedron  may  be  separated  ? 

5.  What  is  the  number  of  sq.  in,  in  the  sum  of  the 
bases  of  the  six  equal  pyramids  into  which  a  12"  cube  may 
be  separated  ?  What  is  the  altitude  of  the  pyramids  in  a 
12"  cube  ?  The  number  of  units  in  its  volume  equals  how 
many  times  the  number  of  units  in  the  base  ?  By  what 
must  the  number  of  units  in  the  surface  of  a  cube  be  mul- 
tiplied to  equal  the  number  of  units  in  the  volume  of  the 
pyramids  to  which  it  is  equal  ? 

6.  The  number  of  units  in  the  volume  of  any  regular 
solid  equals  the  product  of  what  numbers  ? 

7.  If  a  sphere  is  a  regular  polyhedron,  of  what  is  it 
composed  ? 

8.  What  equals  the  number  of  units  in  the  volume  of  a 
sphere  ?  Is  it  necessary  to  use  the.  term  less  radius  in 
giving  a  ruler  for  finding  the  volume  of  a  sphere?  Why 
not? 

9.  Eeview  the  work  on  regular  polygons. 

Before  arriving  at  an  easy  method  of  finding  the  volume  of  a 
sphere,  one  must  know  how  to  find  its  surface. 

The  altitude  and  the  diameter  of  the  base  of  the  cyHnder  D 
are  each  equal  to  the  diameter  of  the  sphere  0.     See  page  204. 


204 


ADVANCED    ARITHMETIC. 


If  a  rectangle  equal  to  the  lateral  surface  of  the  cylinder  D  he  , 
placed  about  the  sphere  0,  it  could,  by  great  skill  in  cutting  and  i 
rearranging,  be  made  to  cover  exactly  the  sphere  0. 


1.  Cut  a  rectangle  equal  to  the  lateral  surface  of  a 
cylinder  whose  altitude  and  diameter  are  each  1  in. 

2.  Cut  a  rectangle  equivalent  to  the  surface  of  a  sphere 
1  in.  in  diameter. 

3.  Cut  a  rectangle  equivalent  to  the  surface  of  a  sphere 
^  in.  in  diameter. 

4.  The  surface  of  the  sphere  -J-  in.  in  diameter  equals 
what  part  of  the  surface  of  the  sphere  1  in.  in  diameter  ? 

5.  Draw  a  rectangle  2  in.  wide  equivalent  to  the  surface 
of  a  sphere  2  in.  in  diameter. 

6.  What  is  the  ratio  of  the  surface  of  a  sphere  2  in.  in 
diameter  to  the  surface  of  a  sphere  1  in.  in  diameter  ?  to 
the  surface  of  a  sphere  4  in.  in  diameter  ? 

7.  If  a  is  the  diameter  of  a  sphere,  what  may  be  the 
dimensions  of  a  rectangle  equal  to  the  surface  of  the  sphere  ? 


ADVANCED    ARITHMETIC. 


205 


The  altitude  and  the  diameter  of  the  cylinder  are  each  equal 
to  the  diameter  of  the  sphere  and  of  the  hemisphere.  The  length 
of  the  cord  which  covers  the  curved  surface  of  the  hemisphere 
equals  the  length  of  the  cord  which  covers  one-half  of  the  lateral 
surface  of  the  cylinder.  Compare  the  lateral  surface  of  the 
cylinder  with  the  surface  of  the  sphere.     See  page  204. 

8.  Observe  spheres  and  cylinders  and  compare  surfaces. 

9.  What  are  the  dimensions  of  a  rectangle  equivalent 
to  a  circle  1  in.  in  diameter  ?     2  in.  in  diameter  ? 

10.  What  are  the  dimensions  of  a  rectangle  equivalent 
to  a  great  circle  of  a  sphere  1  in.  in  diameter  ? 

11.  What  is  the  ratio  of  a  great  circle  of  a  sphere  to  the 
surface  of  the  sphere  ? 

12.  A  great  circle  of  the  earth  equals  what  part  of  its 
surface  ? 

13.  Discover  the  ratios  of  the  surfaces  of  the  solids  in 
the  following  cut  : 


ITii 


"*™^ 


14.    Draw  a  rectangle  equivalent  to  the   surface  of  a 
sphere  1  in.  in  diameter.     Make  rectangle  1  diameter  wide. 


15.  Express  in  diameters  the  dimensions  of  a  rectangle 
equivalent  to  the  surface  of  any  sphere.  What  is  the  ratio 
of  this  rectangle  to  the  square  of  the  diameter  ?  3|  or  ^- 
is  the  ratio  of  the  surface  of  a  sphere  to  what  ? 


206 


ADVANCED   ARITHMETIC. 


16.  What  is  the  name  of  the  equal  solids  into  which 
any  regular  solid  may  be  separated  ?  By  what  must  the 
number  of  units  in  tlie  surface  of  a  regular  solid  be  multi- 
plied to  equal  the  number  of  units  in  the  volume  of  the 
regular  solid  ? 

17.  Give  several  different  methods  of  finding  the  volume 
of  a  sphere. 

18.  Draw  a  rectangular  solid  equal  to  the  volume  of  a 
sphere  1  in.  in  diameter. 

What  are  the  dimensions  of  the  solid  expressed  in 
inches  ?   in  diameters  ? 

19.  What  are  the  dimensions  of  a  rectangular  solid 
equal  to  a  sphere  2  in.  in  diameter.  E^xpress  the  dimen- 
sions in  inches;  in  diameters. 


20.  Draw  the  cube  of  the  diameter  of  a  sphere  1  in.  in 
diameter.  A  solid  3i  d.  long  by  1  d.  wide  by  J  d.  high 
equals  what  part  of  the  cube  of  its  diameter  ? 

21.  Any  sphere  equals  what  part  of  the  cube  of  its 
diameter  ? 

22.  What  is  the  volume  of  spheres  having  the  following 
diameters:  14";  7^" ;  4:2  it  ;  .7  yd.;  2.9ft.;  8f"? 


ADVANCED    AKITHMETIC.  207 

11  ■  14^  _  the  number  of  cubic  inches  in  a  sphere  14  in.  in 

21  diameter. 

23.  Measure  the  diameters  of  many  spherical  bodies  — 
baseballs,  croquet  balls,  drawing  models,  etc.,  and  find  J^ 
of  the  cube  of  their  diameters. 

1.    What  is  the  ratio  of  a  to  b?  c  equals  what  part  of  a? 
d  equals  what  part  oi  b?     Then  what  is  the  ratio  of  c  to 


b 

d?     Spheres   are   to  each  other  as  the  cubes  of  what? 
Why? 

2.  What  is  the  ratio  of  a  sphere  1  in.  in  diameter  to  a 
sphere  2  in.  in  diameter  ?     Why  ? 

3.  What  is  the  ratio  of  a  sphere  2  in.  in  diameter  to  a 
sphere  4  in.  in  diameter  ?  to  a  sphere  3  in.  in  diameter  ?  to 
a  sphere  1  in.  in  diameter  ?  to  a  sphere  -J-  in.  in  diameter  ? 

4.  What  is  the  ratio  of  a  sphere  •§-  ft.  in  diameter  to  a 
sphere  f  ft.  in  diameter  ? 

5.  What  is  the  ratio  of  a  sphere  1  in.  in  diameter  to  a 
2-in.  cube  ?  Ans.  ^\  of  -J  is  the  ratio  of  a  sphere  1  in.  in 
diameter  to  a  2-in.  cube. 

6.  Observe  cylinders,  cones,  cubes,  spheres,  and  other 
solids,  and  give  ratios. 

1.  What  is  the  volume  of  the  largest  sphere  that  can 
be  turned  out  of  a  cubic  foot  of  wood  ? 

2.  What  is  the  surface  of  the  largest  sphere  that  can 
be  turned  out  of  a  cube  2  ft.  long  ? 


208  ADVANCED    ARITHMETIC. 

3.  What  is  the  entire  surface  of  an  18"  cube?  What 
is  the  length  of  its  less  radius  ?  What  is  the  area  of  the 
base  of  one  of  its  six  pyramids  ?  What  is  the  volume  of 
one  of  the  pyramids  ?     What  is  the  volume  of  the  cube  ? 

4.  Give  a  rule  for  finding  the  volume  of  a  cube.  Give 
a  rule  for  finding  the  volume  of  any  regular  solid. 

5.  Compare  the  rule  for  finding  the  area  of  a  regular 
polygon  with  the  rule  for  finding  the  volume  of  a  regular 
solid. 

6.  What  is  the  volume  of  the  largest  cone  that  can  be 
turned  out  of  a  cubic  foot  ? 

7.  What  are  the  relations  of  the  largest  cylinder,  sphere, 
and  cone  that  can  be  turned  out  of  equal  cubes  ? 


8.  Give  two  rules  for  finding  the  volume  of  a  sphere. 

9.  What  is  the  volume  of  a  cube  6  ft.  long  ?     What  is 
its  entire  surface  ? 

10.  The  base  of  a  rectangular  solid  is  a  square  inch. 
What  is  the  altitude  if  its  volume  equals  that  of  a  cylinder 
whose  altitude  is  1  in.  and  the  diameter  of  whose  base  is 
1  in.  ? 

The  base  of  a  rectangular  solid  is  a  square  inch.  What 
is  the  altitude  if  its  volume  equals  that  of  a  sphere  1  in. 
in  diameter  ? 

The  base  of  a  rectangular  solid  is  a  square  inch.  What 
is  the  altitude  if  its  volume  equals  that  of  a  cone  1  in.  in 
altitude  and  the  diameter  of  whose  base  is  1  in.  ? 

What  is  the  relative  height  of  the  three  solids  that  are 
respectively  equal  to  the  cylinder,  the  sphere,  and  the 
cone? 


ADVANCED    ARITHMETIC.  209 

11.  What  is  the  altitude  of  a  rectangular  solid  whose 
base  is  ct^  if  the  volume  of  the  solid  equals  a  cylinder 
whose  altitude  is  a  and  the  diameter  of  whose  base  is  a  ? 

1.  What  is  the  volume  of  a  cylinder  whose  diameter 
and  altitude  are  each  1  ?  6  ?  10  ?  x?  What  is  the  volume 
of  a  sphere  whose  diameter  isl?  6?  10?  x? 

Then  what  is  the  ratio  of  the  volume  of  a  sphere  to  the 
volume  of  a  cylinder  whose  diameter  and  altitude  each 
equal  the  diameter  of  the  sphere  ? 

2.  What  is  the  volume  of  a  cone  whose  greatest  diam- 
eter is  1  and  whose  altitude  is  1  ?  whose  greatest  diameter 
and  altitude  are  6  ?  10  ?  ic  ? 

Then  what  is  the  ratio  of  the  volume  of  a  cone  whose 
altitude  equals  the  diameter  of  its  base  to  the  volume  of  a 
sphere  of  equal  diameter  ? 

3.  What  part  of  the  volume  of  a  wooden  cylinder  whose 
diameter  and  altitude  are  equal  must  be  cut  away  in  order 
to  turn  from  it  the  largest  possible  sphere?  in  order  to 
turn  from  it  the  largest  possible  cone  ? 

4.  Describe  '^  the  three  round  bodies  "  whose  relative 
value  is  expressed  by  the  numbers  1,  2,  3. 

5.  What  part  of  a  cube  equals  the  largest  cylinder  con- 
tained in  it  ?  the  largest  sphere  ?  the  largest  cone  ? 

6.  What  is  the  entire  surface  of  a  cylinder  whose 
diameter  is  1  and  whose  altitude  is  1?  What  is  the  en- 
tire surface  of  a  cylinder  whose -diameter  is  6  and  whose 
altitude  is  6  ?  of  a  cylinder  whose  diameter  and  altitude 
are  10  ?  of  a  cylinder  whose  diameter  and  altitude  are  x  ? 

7.  What  is  the  surface  of  a  sphere  whose  diameter  is  1  ? 
6  ?  10  ?  iP  ? 

8.  What  is  the  ratio  of  the  surface  of  a  sphere  to  the 
curved  surface  of  a  cylinder  whose  diameter  and  altitude 
each  equal  the  diameter  of  a  sphere  ?     What  is  the  ratio 


210  ADVANCED   ARITHMETIC. 

of  the  surface  of  the  spliere  to  the  entire  surface  of  the 
cylinder  ?    the  reciprocal  ratio  ? 

9.    What  is  the  ratio  of  the  curved  surface  of  a  hemi- 
sphere to  the  plane  surface  ? 

10.  What  is  the  ratio  of  the  entire  surface  of  a  sphere 
to  the  entire  surface  of  a  hemisphere  of  equal  diameter  ? 
the  reciprocal  ratio  ? 

11.  What  is  the  ratio  of  the  curved  surface  of  one-half 
a  hemisphere  to  the  sum  of  its  plane  surfaces  ?  Wliat  is 
the  ratio  of  the  entire  surface  of  one-fourth  of  a  sphere  to 
the  entire  surface  of  a  hemisphere  ?  to  the  entire  surface 
of  a  sphere  ? 

12.  What  is  the  ratio  of  the  entire  surface  of  one-eighth 
of  a  sphere  to  a  great  circle  of  that  sphere  ? 

13.  What  is  the  ratio  of  the  surface  of  a  sphere  to  the 
entire  surface  of  a  cube  whose  edge  equals  the  diameter  of 
the  sphere  ? 

14.  What  is  the  ratio  of  the  entire  surface  of  a  hemi- 
sphere to  the  entire  surface  of  the  smallest  square  prism 
that  will  contain  it  ?  What  is  the  ratio  of  the  curved  sur- 
face of  the  hemisphere  to  the  surface  of  such  a  square  prism  ? 

15.  What  is  the  slant  height  of  a  cone  —  diameter  of 
base  1  —  whose  curved  surface  is  double  its  plane  surface  ? 
of  a  cone  whose  plane  surface  equals  i  of  its  curved 
surface  ? 

16.  What  is  the  ratio  of  the  entire  surface  of  each  of  the 
cones  in  the  last  to  the  entire  surface  of  a  hemisphere 
whose  diameter  equals  the  greatest  diameter  of  the  cone  ? 

17.  What  is  the  ratio  of  the  curved  surfaces  of  these 
cones  to  each  other  and  to  the  curved  surface  of  the  hemi- 
sphere ? 

18.  What  are  the  ratios  of  the  curved  surfaces  of  the 
largest  sphere,  cylinder  and  cone  that  can  be  cut  from  equal 
cubes  ? 


ADVANCED   ARITHMETIC. 


211 


100 


200 


300 


600 


•iOO 


500 


Square  Root.  —  1.    What  is  the  ratio  of  each  unit  to  each 
of  the  others  ? 

2.  If  100  is  the  length  of  a,  what  is  the  length  of  each 
of  the  others  ? 

3.  If  a  is  100^  {Read :  If  a  is  the  square  of  1  hundred), 
each  of  the  other  units  is  the  square  of  what  ? 

4.  What  is  the  ratio  of  400^  to  the  square  of  each  of 
the  others  ? 

5.  What  is  the  ratio  of  each  square  to  each  of  the 
others  ? 

6.  What  is  the  ratio  of  700^  to  each  ?    of  800^  ?  of 
900^? 


212 


ADVANCED    ARITHiMETIC. 


7.  4  is  the  ratio  of  tlie  squares  of  which  units  between 
100  and  900  inclusive  ?  9  is  the  ratio  of  the  squares  of 
what  ?   J  is  the  ratio  of  the  squares  of  what  ?  i  ?  16  ?  ^^^  ? 

8.  What  is  the  ratio  each  to  each  between  100'-^  and 
9002  inclusive  ?     What  is  the  ratio  of  200^  to  400^  ?   to 

9.  What  is  the  ratio  of  300^  to  600^  ?  to  900^  ? 
10.    What  is  the  ratio  of  400^  to  200^  ? 


1.  If  10  is  the  length  of  a,  what  is  the  length  of  each 
of  tlie  other  units  ? 

2.  If  a  is  10^,  each  of  the  other  squares  is  the  square  of 
what  ? 

3.  What  is  the  ratio  of  40'^  to  tlie  square  of  each  of  the 
others  ? 


ADVANCED    ARITHMETIC.  213 

4.  What  is  the  ratio  of  20^  to  the  square  of  each  of 
1*ie  others  ?  of  50^  ?  of  60^  ?  of  80^  ?  of  90^  ?  of  70^  ? 

5.  4  is  the  ratio  of  which  squares  ?  9  is  the  ratio 
of  which  squares  ?  :^  is  the  ratio  of  wliich  squares  ?  ^  ? 
iV?  16? 

6.  What  is  the  ratio  of  10^  to  each  between  10^  and 
90^  inclusive  ?    of  20^  to  each  ?    of  30^  ?  of  40^  ? 

1.  What  is  the  ratio  of  100^  to  300^  ?  of  10^  to  30^  ?  of 
1^  to  32  ?  of  .1'  to  .32  ? 

2.  What  is  the  ratio  of  100^  to  400^  ?  of  10^  to  40^  ?  of 
IH042?  of  .12  to  .42? 

3.  What  is  the  ratio  of  100^  to  700^  ?  of  10^  to  70^  ?  of 
V  to  72  ?  of  .12  to  .72  ? 

4.  What  is  the  ratio  of  V  to  6^  ?  to  10^  ?  to  100^  ? 

5.  What  is  the  ratio  of  V  to  9^  ?  to  90^  ?  to  900^  ? 

6.  What  is  the  ratio  of  V  to  7^  ?  to  70^  ?  to  700^  ? 

Write  the  following  in  figures  and  practice  reading : 
1  ten-thousand ;  9  ten-thousand ;  15  ten-thousand ;  71 
ten-thousand ;  13  ten-thousand  ;  7  ten-thousand  8  hundred  ; 
67  ten-thousand  15  hundred ;  49  ten-thousand  17  hundred 
9  tens  ;  63  ten-thousand  22  hundred  4  tens ;  84  ten-thou- 
sand 29  hundred  64  ;  33  ten-thousand  94  hundred  75 ;  20 
ten-thousand  5  hundred  3  tens  ;  91  ten-thousand  7  hundred 
1 ;  70  ten-thousand  27  hundred  82 ;  5  ten-thousand  3  hun- 
dred 7 ;  48  ten-thousand  93  hundred  5  tens  ;  18  ten-thou- 
sand 30  hundred;  12  ten-thousand  9;  3  ten-thousand  3; 
64  ten-thousand  36  hundred  50  ;  60  ten-thousand  2  hun- 
dred 40 ;  6  ten-thousand  3  hundred  7 ;  40  ten-thousand  7 
hundred  1 ;  90  ten-thousand  9  hundred  9 ;  30  ten-thou- 
sand 4  hundred  90  ;  30  ten -thou  sand  30  hundred  30  ;  3  ten- 
thousand  3  hundred  3 ;  45  ten-thousand  1  thousand  10. 


214  ADVANCED   ARITHMETIC. 

16  00  00  is  4001     Read :  16  ten-thousand  is  the  square 
of  4  hundred. 

What  is  the  square  of  300  ?  of  200  ?  500  ?  etc. 

Review  pp.  238,  270,  and  300,  Speer's  "Elementary  Arith- 
metic." 

1.  Can  a  square  be  cut  into  2  equal  squares  ?  into  5 
equal  squares  ?  into  9  ?  into  10  ?  into  15  ? 

2.  When  a  square  is  divided  into  4  equal  squares,  how 
are  its  edges  divided  ?  if  divided  into  9  equal  squares  ? 
etc. 

Ex.  The  edge  of  a  square  that  is  divided  into  4  equal 
squares  is  divided  into  halves. 

3.  What  are  the  different  numbers  of  equal  squares 
between  1  and  100  into  which  any  square  can  be  divided  ? 

4.  What  is  the  least  nuynher  of  equal  squares  of  which 
a  square  can  be  made  ?  Give  all  the  numbers  of  equal 
squares  between  1  and  100  of  which  perfect  squares  can 
be  made. 

5.  What  is  10^?  What  is  the  least  number  of  10^ 
{Read:  squares  of  10)  that  can  be  placed  together  to  make 
a  perfect  square  ?  What  are  the  different  numbers  of  10^ 
between  1  and  100  of  which  perfect  squares  may  be  made  ? 

6.  What  is  100^  ?  What  is  the  least  number  of  100^ 
that  can  be  placed  together  to  make  a  '  perfect  square  ? 
Give  all  of  the  different  numbers  of  100^  between  1  and 
100  that  may  compose  perfect  squares. 

7.  What  is  the  largest  square  composed  of  inch  squares 
that  can  be  made  of  5  sq.  in.  ?  What  is  the  length  of  this 
square  ?     4  is  the  number  of  what  ? 

8.  What  is  the  largest  square  composed  of  100^  (Read: 
squares  of  100)  that  can  be  made  of  5  00  00  (5  ten-thou- 
sand) ?  What  is  the  length  of  this  square  ?  Then  what 
is  V4  00  00  ? 


ADVANCED    ARITHMETIC. 


215 


9.  What  is  the  largest  square  composed  of  10^  that  can 
be  made  of  18  00  ?  What  is  the  length  of  this  square  ? 
What  is  the  Vl600  ? 

10.  What  is  the  largest  square  composed  of  1^  that  can 
be  made  of  S5  ?  of  34  ?  of  10  ?  of  20  ?  of  47  ?  of  75  ?  of 
50  ?     What  is  the  length  of  each  square  ? 

11.  What  is  the  largest  square  composed  of  100^  that 
can  be  made  of  50  00  00?  of  75  23  72  (Bead:  75  ten- 
thousand  23  hundred  72)  ?  of  10  00  00  ?  of  1  00  00  ?  of 
35  73  22  ?  of  99  99  99  ?  of  8  27  31  ?  of  15  12  43  ?  of 
22  22  22  ?     What  is  the  length  of  each  square  ? 

12.  What  is  the  largest  square  that  can  be  made  of  10^ 
(squares  of  10)  contained  in  4  83  ?  in  38  33  ?  in  10  00  ? 
in  27  27?  in  45  45?  What  is  the  length  of  each 
square  ? 

13.  What  is  the  largest  square  that  can  be  made  of  .1^ 
contained  in  .69  ?  in  5.25  ?  in  37.33  ?  What  is  the  length 
of  each  square  ? 

14.  What  is  the  largest  square  of  hundredths  in  .00  38  ? 
in  .12  38  ?  in  4.37  62  ? 


1.  What  is  the  length  of  a  square  containing  81  00  00  ? 
of  a  square  containing  81  00  ?  of  a  square  containing  81  ?  of 
a  square  containing  .81?  of  a  square  containing  .00  81  ? 

2.  What  is  the  length  of  a  square  containing  64  00  00  ? 
49  00  00  ?  16  00  00  ?  64  00  ?  .64  ?  49  00  ?  49  ?  .49  ? 


216 


ADVANCED    ARITHMETIC. 


3.  What  is  the  length  of  a  square  containing  25  00  00  ? 
25  00  ?  25  ?  .25  ?  .00  25  ?  .00  00  25  ? 

Write  and  memorize  the  squares  of  all  the  different 
numbers  of  hundreds  from  100  to  900  inclusive.  Image 
them  as  divided  into  1001  Ex. :  In  the  300^  I  see  9  squares 
of  100,  or  9  00  00  {Read:  9  ten-thousand).  Write  and 
memorize  the  squares  of  all  the  different  numbers  of  tens 
from  1  ten  to  90  inclusive.  Image  them  as  divided  into 
101     How  many  hundred  in  each  ? 

Write  and  memorize  the  squares  of  all  the  different 
numbers  of  tenths  from  .1  to  .9  inclusive.  How  many  .1^ 
in  each  ?     How  many  hundredths  in  each  ? 


ADVANCED   AKITHMETIC. 


217 


100^  =  1  00  00 
200^  =  4  00  00 
3002  ^  9  00  00 
400^  -  16  00  00 
500^  =  25  00  00 
6002  =  36  00  00 
700^  =  49  00  00 
8002  ^  04  Q()  OQ 

9002  =  81  00  00 
1002  = 
102  = 

12  = 


102  =    1  00 
202  =    4  00 
302  =    9  00 
402  =  16  00 
502  ^  25  00 
602  ^  3(3  00 
702  _  49  00 
802  ^  g4  00 
902  ^  8^  00 
1  00  00. 
1  00. 
1. 


12=    1 

22=    4 

32=    9 

42  =  16 

52  =  25 

62  =  36 

72  =  49 

82  =  64 

92  =  81 

4002  =  16  00  00. 

402=    16  00. 

.42  =      .16. 


.12  =  .01. 
.22  =  .04. 
.32  =  .09. 
.42  =  .16. 
.52  =  .25. 
.62  =  .36. 
.72  =  .49. 
.82  =  .64. 
.92  =  .81. 


Write  the  above  from  memory.  Begin  at  the  top  and 
write  to  the  bottom ;  reverse.  Begin  at  left  and  write  to 
right ;  reverse.     Give  these  equations  at  random. 


1.  What  is  the  largest  square  that  can  be  made  of  5' 
(5  ■  12)  ?  of  5  00  (5  •  102)  ?  of  5  00  00  (5  '  IOO2)  ? 

2.  What  is  the  edge  or  root  of  the  largest  square  in  5 
composed  of  I2  ?  of  the  largest  square  in  5  00  composed  of 
102  ?  the  largest  in  5  00  00  composed  of  IOO2  ?  the  largest 
in  .05  composed  of  .I2  ?  the  largest  in  .00  05  composed  of 
.012? 

3.  What  is  the  largest  square  of  ones  in  79  ?  of  tens 
in  79  00  ?  of  hundreds  in  79  00  00  ?  of  tenths  in  .79  ?  of 
hundredths  in  .00  79  ? 


218 


ADVANCED    ARITHMETIC. 


4.  What  is  the  largest  square  of  tens  in  5  25  ?  its 
root  ?  remainder  ?  Ans.  400  is  the  largest  square  of  tens 
in  525 ;  its  root  is  20 ;  the  remainder  is  125. 

5.  What  is  the  largest  square  of  tens,  its  root,  and  the 
remainder  in  each  of  the  following  :  6  24  ;  11  62  ;  16  79 ; 
75  25  ? 

6.  What  is  the  largest  square  of  hundreds,  its  root,  and 
the  remainder  in  each  of  the  following  :  6  24  00 ;  11  62  00 ; 
16  79  00  ;  92  16  00  ;  75  25  00  ? 

7.  What  is  the  largest  square  of  hundredths,  its  root, 
and  the  remainder  in  .11  23  ?  Ans.  .09  is  the  largest 
square  of  hundredths,  .3  is  its  root,  and  .02  23  is  the 
remainder. 

8.  What  is  the  largest  square  of  hundredths,  its  root, 
and  the  remainder  in  each  of  the  following  :  .06  24  ;  .11  62  ; 
.16  79  ;  .92  16  ;  .75  25  ;  .40  ;  .50  ? 

9.  What  is  the  largest  square  of  ones,  its  root,  and  the 
remainder,  expressed  in  hundredths,  in  each  of  the  follow- 
ing: 10;  li;  12.38;  6.7;  3f ;  4.2?     Ux.  9  is  the  largest 

square  of  ones  in  10; 
its  root  is  3,  and  the  re- 
mainder is  1.00  (i§§). 

10.  What  is  the  larg- 
est square  of  hundredths 
ill  .83  ?  in  .80  ?  in  .8  ? 
in  .66|  or  f  ?  in  .50  or 
.5  ?  in  .75  or  f  ?  in  .12  ? 
in  .121?  in  .125?  in^? 
in  .375  or  f  ? 

11.    In    the    above, 

what  is  the  edge  of  each  square  and  what  is  the  remainder  ? 
12.    What  is  the  largest  square  of  ten-thousandths  in 

.0038?  in  .0082?  in  .004  ?  in  .006  ?  in  .0016  ?  in  .0076  ? 

ini?   in  I?    mi? 


ADVANCED    ARITHMETIC. 


219 


13.  What  is  the  largest  square  of  hundreds  in  576  ? 
What  is  the  length  of  the  edge  of  the  square  400  ?  To 
how  many  sides  of  the  square  400  must  the  176  be  added 
to  preserve  the  form  of  a  square  ?  What  is  the  length  of 
the  sum  of  two  edges  of  the  square  400  ?  If  the  area  to 
be  added  is  176,  what  is  the  approximate  width  of  the 
additions  ? 

1.  Show  by  a  drawing  the  form  of  the  difference  of 
two  squares.  Straighten  the  X-shaped  difference  into  a 
rectangle  without  altering  its  width.  The  length  of  the 
rectangle  equals  how  much  more  than  twice  the  length  of 
the  smaller  square  ? 

2.  What  is  the  largest  square  of  hundreds,  its  root, 
and  the  remainder  in  625  ?  Conceive  the  remainder  as  a 
rectangle.  What  is  its  area?  Approximate  its  length. 
If  its  length  were  40,  what  would  be  its  approximate  width  ? 
If  the  length  of  this  rectangle  were  45  and  its  width  5, 
what  would  its  area  be  ?  Then  225  is  the  difference 
between   what   square   and  20^? 


625(25 
4 


45 


225 
225 


}ii) 


What  is,  therefore,  the  V625  ?  of  625  half-inch  squares  ? 
of  6.25  ?  of  625  equal  squares  of  any  kind  ? 

What  is  the  square  root  of  625  sq.  ft.  ?  of  625  sq.  yd.  ? 
of  625  sq.  rd.  ?  of  6.25  ? 


220 


ADVANCED    ARITHMETIC. 


3.    What  is  the  edge  of  the  largest  square  of  hundreds 
in  529  ? 


100 


20 


43 


529  (23 
400 
129 
129 


40 


What  is  the  largest  square  of  hundreds  in  500  ?  Show 
the  400  in  the  diagram. 

What  is  the  length  of  two  sides  of  the  square  to  which 
the  129  is  to  be  added  ? 

If  129  is  the  number  of  units  in  the  area  of  the  addition, 
and  40  is  nearly  the  number  in  the  length,  what  is  the 
number  in  the  width  ? 

Then  129  is  the  difference  between  the  square  of  20  and 
the  square  of  what  ?  What  is,  therefore,  V529 '-  o^  ^^9 
sq.  ft.  ?  of  5.29  ?  of  .0529  ? 

Mentally  picture  the  work  done  in  finding  the  edge  of 
the  square  529. 

What  is  the  edge  of  the  square  484  ? 

4.   What  is  the  length  of  the  edge  of  the  square  of  the 
hundreds  in  the  square  1764  ? 

What  is  the  length  of  the  square  1600  ? 


ADVANCED    ARITHMETIC. 


221 


What  is  the  sum  of  two  edges  of  the  square  1600  ? 

If  80  is  nearly  the  number  of  units  in  the  length  of  a 
rectangle  and  164  is  the  number  of  units  in  its  area,  what 
do  you  think  is  its  width  ? 

What  is  the  length  of  the  square  1764  ?  What  is  the 
square  root  of  1764  ? 

5.    What  is  the  square  root  of  1024  ? 


f 


WM/^W/MWMWWfWmwA 


30 


+ 


2 


62 


10  24  (32 

9 
124 
124 


6.  Find  the  square  root  of  53  29  ;  8  41 ;  72  25  ;  98  01  ; 
10.89;  14.44;  .28  09;  .29  16;  .20  25;  .8649;  .72  8. 

7.  What  is  the  edge  of  the  square  of  the  hundreds 
in  552? 


5  52(23 
4 


43 


152 

129 

23 


8.    What  is  the  edge  of  the  square  5  52.25  ? 


222 


ADVANCED    ARITHMETIC. 


552.25  (23.5 
4 


43 


152 
129 


1.    What  is  the  square  root  of  552.25  ? 

What  is  the  largest  square  of  hundreds  in  552  ?     What 
is  its  root  ?     What  is  the  remainder  ? 

What  is  the  length  of  the  sum   of 
two  sides  of  the  square  ? 

If  152  is  the  number  of  units  in  the 
sum  of  the  rectangles  to  be  added,  and 
40  is  nearly  the  number  in  the  sum  of 
their  lengths,  what  is  the  number  in  the 
width  ? 
If  3  is  the  number  in  the  width,  what  is  the  entire 
number  in  the  length  ?     Why  add  3  to  40  before  multiply- 
ing by  3  ? 

What  is  now  the  number  of  tenths  in  the  sum  of  two 
sides  of  the  square  ? 


46.5 


23.25 
23.25 


ADVAJSCED   ARITHMETIC. 


223 


If  2325  is  the  number  of  hundredths  in  the  sum  of  the 
rectangles  to  be  added,  and  460  is  nearly  the  number  of 
tenths  in  the  sum  of  their  lengths,  what  is  the  number 
of  tenths  in  the  width  ? 

465  tenths  is  the  sum  of  what  ?     .5  of  46.5  equals  what  ? 

.-.552.25  =  ? 

What  is  the  square  root  of  5.52 '25  ?  of  5  52  25  ? 
2.   Find  Vll  90  25 :  Vl  65.49. 


1.  What  is  the  ratio  of  4  to  9  ?     What  is  the  ratio  of 
an  edge  of  the  4  to  an  edge  of  the  9  ? 

Then  does  Vj  =  f  ? 

2.  What  is  V^;  V||;  V^;  V^;  V:^;  V^? 


1.  Divide  a  square  into  tenths  and  make  the  equal  parts 
squares  if  you  can. 

Can  a  square  be  divided  into  tenths  making  each  tenth 
a  square  ? 

Is  .1  by  .1  of  a  square  square  ? 

What  is  the  name  of  such  a  square  ? 

What  is  the  length  of  the  square  .01  ? 

A  square  .1  by  .1  equals  what  part  of  V? 

A  square  .2  by  .2  equals  how  many  hundredths  ?  .7  by 
.7  ?  .5  by  .5  ? 


224 


ADVANCED    ARITHMETIC. 


Into  what  must  tenths  be  changed  before  the  square  root 
can  be  found  ? 

2.    What  is  the  square  root  of  .5  ? 

.5  equals  how  many  hundredths  ? 

What  is  the  largest  square  of  hundredths  in  .50  ?    What 
is  the  square  root  of  .49  ?    of  .50  ? 


What  is  the  square  root  of  .7  ?  of  .15  ?  of  .3  ?  of  1.1  ? 
of  3.3  ? 

3.  Can  you  find  the  square  root  of  .007  ? 

Why  not  ?     .007  equals  how  many  ten-thousandths  ? 
What  is  the  largest  square  of  ten-thousandths  in  .0070  ? 
What  is  the  root  or  edge  of  .00  64  ? 

Ans.    V.OO  7=  V.OO  70  =  .08 +. 

4.  Show  the  places  that  the  squares  of  hundreds,  tens, 
ones,  tenths,  hundredths,  and  thousandths  respectively 
occupy. 

5.  How  many  figures  in  the  square  root  of  each  of  the 
following  :  9  25 ;  84  32  ;  7  85.95  ;  8  76.5 ;  42.73  0  ? 

6.  What  is  the  largest  square  of  ones  in  5  ?  of  hun- 
dredths in  .05  ?  of  ten-thousandths  in  .0005  ?  What  is 
the  approximate  square  root  of  each? 


ADVANCED    ARITHMETIC. 


225 


7.    What  is  the  square  root  of  2  carried  to  three  decimal 
places  ? 

2.00  00  00  (1.414 
1 
24 


100 
96 


281 
2824 


400 

281 


11900 
11296 


8.    Carry  each  to  two  decimal  places  : 

V3;  V23;  V^J;  Vt^;  V:05|. 


226 


ADVANCED   ARITHMETIC. 


1.  Eeview  p.  166. 

2.  Show  me  the  squares  a,  h,  c. 

3.  What  does  the  sum  of  the  squares  a  and  h  equal  ? 

4.  The  sum  of  a  and  what  equals  c  ?     The  sum  of  h 
and  what  equals  c  ? 

5.  c  is  how  much  larger  than  b  ? 

6.  If  c  and  a  are*  given,  how  may  the  edge  of  h  be 
found  ? 

7.  If  0^  and  h  are  given,  how  may  c  be  found  ?   the 
edge  of  c  ? 

8.  If  an  edge  of  a  and  of  b  is  given,  how  may  the  edge 
of  c  be  found  ? 

9.  The  square  of  the  hypothenuse  of  a  right  triangle 
equals  the  sum  of  what  ? 


10.  The  sum  of  o^  and  m^  equals  the  square  of  what  ? 

11.  The  square  of  h  is  how  much  more  than  the  square 
of  (?? 


ADVANCED   ARITHMETIC.  227 


12.  Vq^  +  m'^  =  what  ? 

13.  VA^  -  m2  =  what  ? 

14.  If  ^  and  o  are  given,  how  may  m  be  found  ? 

1.  How  long  a  rafter  is  required  in  building  a  house 
24  ft.  wide  if  the  apex  of  the  roof  is  above  the  middle  line 
of  the  house  and  is  9  ft.  higher  than  the  eaves  ? 

2.  How  long  is  the  string  of  a  kite  that  is  ^  mi.  above 
the  earth  and  directly  above  a  point  ^  mi.  away  from  the 
flyer  ? 

3.  What  is  the  slant  height  of  a  cone  whose  altitude 
is  4  in.  and  the  diameter  of  whose  base  is  2  in.  ? 

4.  How  many  rods  is  it  from  one  corner  of  a  square 
40-acre  field  to  the  opposite  corner  ?  How  much  does  a 
pedestrian  save  in  walking  ^'across  lots ''  when  going  from 
one  corner  to  the  other  ? 

5.  How  long  is  the  diagonal  of  each  wall  of  the  school- 
room ?  of  the  ceiling  ? 

6.  What  is  the  ratio  of  the  diagonal  of  any  square  to 
its  side  ?  of  the  side  to  the  diagonal  ? 

7.  Show  that  V|  =  ^  V2  by  inspecting  the  diagram ; 
by  calculating  each. 

8.  What  is  the  diagonal  of  a 
square  10-acre  field  ? 

9.  Make  a  short  rule  for  find- 
ing the  diagonal  of  any  square 
from  its  side.  For  finding  a  side 
from  a  diagonal.  If  a  side  is  s, 
what  is  the  diagonal  ?  If  the 
diagonal  is  d,  what  is  the  side  ? 


Ans.    sV2;  -V2. 

10.    What  is  the  diagonal  of  a  rectangle  3  by  4?  6  by  8  ? 
12  by  16  ?     If  the  altitude  of  a  rectangle  equals  f  of  the 


^m^^^^^^^^M 

t   ^ 

■"1 

1 

,^ 

1 

t 

\ 

1 

V 

\v' 

228  ADVANCED    ARITHMETIC. 

base,  what  is  the  ratio  of  the  diagonal  to  the  base  ?  to  the 
altitude  ? 

11.  Find  the  diagonals  of  the  different  faces  of  a  rectan- 
gular prism  3x5x8.  Find  the  diagonal  of  the  prism 
itself. 

12.  What  is  the  distance  from  the  upper  N.  E.  corner  of 
the  schoolroom  to  the  lower  S.  W.  corner  ? 

13.  What  is  the  ratio  of  the  diagonal  of  a  cube  to  each 
of  its  edges  ? 

14.  Calculate  the  lateral  sides  of  an  isosceles  triangle 
whose  altitude  is  8  in.  and  whose  base  is  5  in. 

15.  What  is  the  slant  height  of  the  largest  cone  that 
can  be  cut  from  a  cubic  foot  ? 

16.  Find  the  slant  height  of  the  largest  cone  that  can 
be  cut  from  a  cube  whose  edge  is  a. 

17.  What  is  the  slant  height  of  a  square  pyramid  the 
area  of  whose  base  is  4  sq.  in.  and  whose  altitude  is  4  in.  ? 

18.  What  is  the  length  of  each  edge  of  such  a 
pyramid  ? 

19.  What  is  the  length  of  a  ladder  that  just  reaches  a 
window  40  ft.  from  the  ground,  the  foot  of  the  ladder  being 
18  ft.  distant  from  the  wall  ? 

20.  What  is  the  area  of  a  square  whose  diagonal  is  (Z  ? 

21.  What  is  the  diagonal  of  a  square  whose  area  is  a  ? 

22.  Show  that  VoT-  V2  =  V2a  by  the  last  diagram. 

23.  If  the  diagonal  of  a  rectangle  is  d  and  its  altitude  a, 
what  is  its  base  ?  Ans.    b  =  Vc?^  —  a\ 

24.  If  the  hypothenuse  of  a  right  triangle  is  h  and  its 
base  is  h,  what  is  the  perpendicular  ?  ._ 

Ans.  p  =  VA^  —  h^. 

25.  A  is  25  mi.  east  and  B  is  30  mi.  south  of  C.  How 
far  apart  are  A  and  B  ? 

26.  What  is  an  edge  of  the  square  that  equals  the 
difference  between  8^  and  5^? 


ADVANCED   ARITHMETIC.  229 

27.  A  house  is  32  ft.  wide.  The  roof  measures  46  ft. 
from  eave  to  eave.  How  high  above  the  eaves  is  the 
highest  point  of  the  roof  ? 

28.  How  far  is  it  on  a  level  through  a  hole  100  ft.  deep 
but  1450  ft.  long  on  a  slant  from  bottom  to  top  ? 

29.  If  the  side  of  an  equilateral  triangle  is  1,  what  is  its 
altitude  ?     If  the  side  is  s,  what  is  the  altitude  ? 

30.  If  the  side  of  a  rhombus  is  1  and  its  shorter  diagonal 
is  1,  what  is  the  longer  diagonal  ? 

31.  What  is  the  diagonal  of  a  regular  hexagon  whose 
side  is  1  ?  whose  side  is  s?  Ans.    s  •  V3. 

32.  What  is  the  less  radius  of  a  regular  hexagon  whose 
side  is  1  ?  whose  side  is  5  ? 

33.  What  is  the  altitude  of  an  isosceles  triangle  whose 
base  is  60  and  whose  equal  sides  are  each  70  ? 

34.  What  is  the  shorter  diagonal  of  a  rhombus  whose 
longer  diagonal  is  60  and  whose  sides  are  40  ? 

35.  What  is  the  longer  diagonal  of  a  rhombus  whose 
shorter  diagonal  is  80  and  whose  sides  are  100  ? 

36.  One  side  of  a  rectangle  is  60  ft.;  its  diagonal  is  75 
ft.     What  is  the  other  side  of  the  rectangle  ? 

37.  What  is  the  ratio  of  the  greater  radius  of  a  square 
to  an  edge  ?  to  the  less  radius  ? 

Of  the  greater  radius  of  a  cube  to  an  edge  ?  of  the  less 
radius  ? 

38.  Let    r   and  s  represent   the  areas  of   two  squares. 

Represent  the  ratio  of  their  edges.  Ans.     —=• 

■\s 

1.  Draw  two  equal  or  unequal  lines.  Draw  a  line 
equal  to  their  sum.  Draw  the  square  of  each  of  these 
three  lines.  Is  the  square  of  the  sum  of  the  edges  of  two 
squares  equal  to  the  sum  of  the  squares  ?  Test  this 
principle  with  other  lines. 


230 


ADVANCED    ARITHMETIC. 


2.  What  must  be  added  to  the  sum  of  two  squares  to 
equal  the  square  of  the  sum  of  their  edges  ?  Show  by 
diagram. 

3.  What  must  be  added  to  the  sum  of  V  and  5^  to  make 
(7  +  5)2?  What  equals  (8  +  7)2?  (2  +  6)^?  (11 +4)^? 
(6  +  9)2?  (5  +  10)2?   (3  4-12)2?  (a  +  bf?  (x  +  yY"! 

4.  What  equals  {x  +  1)^  ?   {a  +  1)^  ?  (4  +  1)^  ?  (8  +  1)^  ? 

5.  What  equals  the  difference  between  two  squares 
whose  roots  are  consecutive  numbers? 

6.  6^  =  5^  +  11.  8^=72  + what?  9^  =  8^  + what? 
102  =  92  +  what  ?     {x''  +  1^)  =  a;2  +  wj^at  9 

7.  21^  =  400  +  what  ?  51^  =  2500  +  what  ?  91^  = 
what  ?     41^  =  what  ? 

8.  What  equals  101^?  201^?  SOP?  99^?  19^?  29^? 
39=^?  (x-lf? 

9.  Does  the  difference  between  two  squares  equal  the 
square  of  the  difference  between  their  roots  ? 

10.  Make  a  diagram  to  show  how  much  must  be  taken 
from  the  sum  of  two  squares  to  leave  a  remainder  that  is 
equal  to  the  square  of  the  difference  between  their  roots. 


11.    What  must  be  taken  from  a^  +  h'^  to  leave  (a  —  bf 
added  to  (a  -  by  to  equal  a^'-hb^? 


ADVANCED    ARITHMETIC. 


231 


12.  What    equals   the   diiference   between   a?-  +  h^  and 

13.  How  much  must  be  added  to  {a  —  h^  to  make 
(a  +  ^)'? 

14.  What  equals  the  square  of  (7-4)?  of  (8  -  5)?  of 
(12-9)?  of  (13-10)? 

15.  Cut  a  figure  equal  to  the  difference  between  two 
squares.  Cut  an  equivalent  rectangle  whose  base  equals 
the  sum  of  the  square  roots  or*  edges.  What  is  the  altitude 
of  this  rectangle  ? 

16.  Indicate  the  difference  between  a?  and  h^.  Indicate 
area  of  the  rectangle  in  question  15.     Make  an  equation. 

Ans.    a}  -h''={a  +  h)-  {a-  h). 

17.  What  equals  252-52?  32^-82?  18^-122?  722-28^? 

18.  What  must  be  added  to  3^  to  equal  13^?  to  equal 
272?  122?  3329  729 

19.  What  equals  the  side  of  a  square  equivalent  to  a 
rectangle  whose  base  is  b  and  altitude  a  ? 

To  find  the  side  of  such  a  square,  make  CD  equal  the 
sum  of  a  and  h.     Describe  a  semicircumference  on  CD. 


From  E  erect  a  perpendicular  terminating  in  the  curve. 
This  is  a  side  of  such  a  square. 

20.  What  is  the  edge  of  a  square  field  equal  to  a  rec- 
tangular field  10  rd.  by  16  rd.  ? 

An  arc  is  any  part  of  the  circumference  of  a  circle. 

A  chord  is  a  straight  line  joining  any  two  points  in  the 
circumference  of  a  circle. 


232  ADVANCED    ARITHMETIC. 

An  ordinate  is  half  a  chord  ;  the  distance  from  any  point 
in  the  diameter  of  a  circle  to  a  point  on  the  circumference 
directly  opposite  is  an  ordinate. 

21.  What  is  the  value  of  an  ordinate  the  foot  of  which 
is  5  ft.  from  the  center  of  a  circle  whose  radius  is  10  ft.  ? 
In  this  circle,  what  is  the  value  of  a  chord  4  ft.  from  the 
center  ?  3  ft.  from  the  center  ?  1  ft.  ? 

22.  In  a  circle  whose  radius  is  6  in.,  what  is  the  value 
of  a  chord  that  is  2  in.  from  the  center  ?  of  a  chord  whose 
greatest  distance  from  the  arc  which  it  subtends  is  2  in.  ? 
1  in.? 

23.  In  a  circle  whose  radius  is  5  in.,  what  is  the  value 
of  a  chord  whose  grjeatest  distance  from  the  arc  which  it 
subtends  is  1  in.  ?  2  in.  ?  3  in.  ?  4  in.  ? 

24.  In  a  circle  whose  radius  is  r,  what  is  the  value  of  an 
ordinate  whose  foot  is  x  in.  from  the  circumference  ? 

25.  In  a  circle  whose  radius  is  10  ft.,  calculate  the  dis- 
tance of  the  circumference  from  points  on  tlie  diameter  at 
intervals  of  2  ft.  Calculate  the  departure  of  the  circum- 
ference from  a  straight  line  touching  the  circle  and  parallel 
to  the  diameter. 

26.  Wliat  is  the  ratio  of  the  altitude  of  an  equilateral 
triangle  to  a  side  ?  If  the  side  is  1,  what  is  the  altitude  ? 
If  the  altitude  is  1,  what  is  a  side  ?  If  a  side  is  1,  what  is 
the  area  ?  if  the  side  is  s  ?  If  the  area  is  1,  what  is  the 
square  of  a  side  ?     What  is  a  side  ?  VTOO 

Ans.  to  last,    s  = — ;= — 
V43 

27.  If  the  area  of  an  equilateral  triangle  is  1  acre,  what 
is  the  length  of  each  side  ?  the  altitude  ? 

.         VlOO  160      ^,         •  ,      ,    .      ..       o  V43~ 

Ans.    — -== =  the  number  or  rods  m  side.    J  — ; = 

V43  VlOO 

number  of  rods  in  altitude. 

28.  If   the    side   of   a  hexagon   is  1,  what  is  the  less 


ADVANCED    ARITHMETIC.  233 

radius  ?     If  the  less  radius  is  1,  what  is  a  side  ?     If  a  side 
is  1,  what  is  the  area  ?     If  the  area  is  1,  what  is  a  side  ? 

Vioo" 


Ans.    s  = 


V6.43 

What  is  the  ratio  of  the  less  radius  to  a  diagonal  ?  of  a 
greater  radius  to  a  diagonal  ?    of  a  diagonal  to  a  side  ? 

29.  What  is  the  ratio  of  a  hexagon  to  an  equilateral  tri- 
angle if  the  ratio  of  the  side  of  the  hexagon  to  the  side  of 
the  equilateral  triangle  is  -J  ? 

30.  What  is  the  length  of  the  smallest  square  from 
which  a  regular  octagon  may  be  cut  whose  side  is  1  ? 
What  is  the  area  of  this  square  ?  What  is  the  area  of 
each  of  the  four  right  triangles  that  must  be  subtracted 
from  the  square  to  make  the  regular  octagon  ?  Then  what 
is  the  area  of  the  regular  octagon  whose  side  is  1  ?  What 
is  the  ratio  of  the  area  of  the  regular  octagon  to  the  area 
of  the  square  ?  of  the  area  of  the  square  to  the  area  of 
the  regular  octagon  ? 

31.  If  the  side  of  the  square  is  1,  what  is  the  side  of 
the  largest  regular  octagon  that  may  be  made  from  it  ? 

100 

Ans,    .  =  — . 

If  a  falling  body  moves  16.  ft.  the  first  second,  3  times 
16  ft.  the  second  second,  5  times  16  ft.  the  third  second, 
and  7  times  16  ft.  the  fourth  second,  how  far  does  it  fall 
in  4  sec.  ?  How  far  does  it  fall  in  5  sec.  ?  in  8  sec.  ?  in 
50  sec.  ?  in  1  min.  ?  in  a?  sec.  ? 

At  this  rate,  how  long  does  it  take  a  body  to  fall  144  ft.  ? 
576  ft.  ?  1000  ft.  ?  1  mi.  ?  x  ft.  ? 

What  do  you  notice  about  the  sum  of  all  the  odd  numbers 
from  1  to  19  inclusive  ?  About  the  sum  of  all  the  odd 
numbers  from  1  to  any  other  odd  number? 

If  I  cut  a  hole  an  inch  square  in  a  cardboard  and  hold  it 
J  ft.  from  an  electric  light,  how  much  of  a  surface  parallel 


234  ADVANCED    ARITHMETIC. 

to  the  card  and  2  ft.  from  the  light  will  be  illuminated  ? 
3  ft.  from  the  light  ?  7  ft.  ?  10  ft.  ?  x  ft.  ?  Then  how  does 
the  intensity  of  the  light  upon  1  sq.  in.  of  surface  in  any 
of  these  cases  compare  with  that  upon  1  sq.  in.  in  any  of 
the  other  cases  ?  Is  the  ratio  of  surfaces  illuminated  the 
same  as  the  ratio  of  distances  ? 

How  far  from  a  candle  is  the  illumination^Ilg^  as  great  as 
it  is  1  ft.  from  the  candle  ?  yi^  as  great  ?  ^i^  as  great  ? 
1  as  great  ? 

Similar  Figures.  —  1.  Compare  the  surface  of  a  1-in.  cube 
with  the  surface  of  a  2-in.  cube,  of  a  3-in.  cube,  of  a  4-in. 
cube,  and  of  a  5-in.  cube.     Compare  each  with  each. 

2.  Compare  the  surface  of  a  cube  whose  edge  is  x  with 
the  surface  of  a  cube  whose  edge  is  y. 

3.  What  is  the  edge  of  a  cube  whose  surface  is  ic  ? 

4.  Compare  the  surfaces  of  a  1-in.  sphere,  a  2-in.  sphere, 
a  3-in.  sphere,  a  4-in.  sphere,  and  a  5-in.  sphere,  each  with 
each. 

5.  What  is  the  ratio  of  a  sphere  whose  diameter  is  x 
to  a  sphere  whose  diameter  is  y?  What  is  the  ratio  of 
their  surfaces  ? 

6.  Compare  the  lateral  surfaces  of  two  equilateral  tri- 
angular prisms  whose  edges  at  the  base  are  6  in.  and  2  in. 
respectively,  and  whose  altitudes  are  in  the  same  ratio  as 
the  edges  of  their  bases.  Are  the  bases  in  the  same  ratio 
as  the  lateral  surfaces  ?  as  the  edges  of  their  bases  ? 

7.  Two  surfaces  are  similar  when  their  corresponding 
parts  are  in  the  same  ratio  or  when  their  corresponding 
lines  are  in  the  same  ratio.  Is  the  ratio  of  the  correspond- 
ing lines  equal  to  the  ratio  of  the  corresponding  parts  ? 

8.  Draw  a  triangle  whose  sides  are  3,  4,  and  5.  Draw 
one  similar  to  it  but  not  equivalent.  What  is  the  ratio  of 
their  corresponding  parts  ?  of  their  corresponding  lines  ? 


ADVANCED    ARITHMETIC. 


235 


9.    Similar  surfaces  are  to  each  other  as  the  squares  of 
corresponding  lines.     What  is  the  ratio  of  similar  solids  ? 


Distance  and  Time.  —  1.    In  what  direction  is  the  earth 
turning  ?     Show  by  gesture. 

2.  What  is  it  that  causes  the  earth's  surface  to  pass 
under  the  sun's  rays  ? 

3.  How  much  of  the  earth's  surface  is  under  the  sun's 
rays  now  ? 

4.  In  what  time  does  the  now  lighted  surface  pass 
into  the  shadow  of  the  earth  ? 

5.  To  a  man  in  a  balloon  5  m.i.  above  Chicago,  would 


236 


ADVANCED    ARITHMETIC. 


the  sun  indicate  a  passage  of  time  if  the  balloon  did  not 
move  with  the  turning  earth  ? 

6.    If  the  man  were  directly  over   Chicago,  on   what 
horizon  would  he  see  Chicago  disappear  ? 

What  time  would  elapse  until  it  would  be  directly  under 
him  again  ? 


How  long  a  time  between  its  disappearance  and  reappear- 


ance 


In  what  direction  would  he  look  to  see  it  reappear  ? 

If  the  balloon  stopped  turning  with  the  earth  at  six 
o'clock  in  the  morning,  near  what  horizon  would  the  sun 
be  all  the  time  ? 


ADVANCED    ARITHMETIC. 


237 


-^1  ^1 


7.  What  grand  divisions  and 
large  bodies  of  water  would  pass 
under  the  man  in  the  balloon  before 
he  would  see  Chicago  again  ? 

8.  The  tops  of  what  mountains 
would  pass  near  him  ? 

9.  Which  would  pass  him  first, 
Erie,  N.  Y.,  or  Eome,  Italy  ? 

10.  Which  of  these  cities  has  sun- 
rise first  ?     Why  ? 

11.  How  can  any  particular  point 
in  the  sky  be  located  ? 

12.  At  midnight  Chicago  is  under 
a  particular  star ;  at  6  a.m.  it  is  un- 
der a  star  that  is  how  far  from  the 
other?  What  place  or  places  will 
then  be  under  the  first  star  ? 

13.  If  we  wish  to  locate  points  in 
the  sky  that  will  be  upon  our  merid- 
ian at  the  even  hours,  how  far  apart 
must  we  place  them  ? 

14.  If  Chicago  is  under  one  of 
these  points  at  8  p.m.,  what  time 
will  it  be  at  San  Francisco  when  San 
Francisco  passes  under  this  point  ? 

See  p.  243. 

1.  How  many  90  degrees  in  the 
circumference  of  the  earth  ? 

2.  In  what  time  does  the  earth 
turn  90  degrees  ? 

3.  How  many  15  degrees  in  90 
degrees  ?  In  what  time  does  the 
earth  turn  15  degrees  ?  30  degrees  ? 
1  degree?  30'?  15'?  60"?  15"? 


238  ADVANCED   ARITHMETIC. 

4.  How  far  does  the  earth  turn  in  6  hr.  ?  in  1  hr.  ? 
in  4  min.  ?    in  1  min.  ?    in  60  sec.  ?   in  4  sec.  ? 

5.  The  arc  AB  is  1  degree  in  length,  or  ^^^  of  the 
circumference  of  the  circle,  and  it  represents  the  distance 
that  the  earth  turns  in  4  min. 

6.  Show  the  part  of  the  arc  AB  that  represents  the 
distance  the  earth  turns  in  1  minute.  One  degree  equals 
how  many  minutes  of  distance  ? 

7.  The  earth  turns  how  many  minutes  of  distance  — 
longitude  —  in  1  minute  of  time  ? 

8.  In  how  many  seconds  of  time  does  the  earth  turn 
from  Mto  0?  from  Dto  U? 

9.  In  how  many  seconds  does  the  earth  turn  1  min.  of 
distance  ? 

10.  In  4  sec.  of  time  the  earth  turns  how  many  seconds 
of  distance  ? 

11.  In  any  time,  what  is  the  ratio  of  the  number  of 
hours  to  the  number  of  degrees  which  the  earth  turns  ?  of 
the  number  of  minutes  of  time  to  the  number  of  degrees  ? 

of  minutes  of  time  to  minutes  of  dis- 
tance ?  of  seconds  of  time  to  minutes 
of  distance  ?  of  seconds  of  time  to 
seconds  of  distance  ? 

12.   Observe  the  diagram  carefully. 
Practice  mentally  picturing  it.  Draw- 
ing the  diagram  from  memory  will 
aid  greatly  in  retaining  the  mental  picture. 

13.  Give  the  converse  of  the  above  ratios.  Ex. :  15  is 
the  ratio  of  the  number  of  degrees  to  the  number  of  hours. 

14.  Make  sentences  like  this  :  If  3  is  the  number  of 
hours,  45  is  the  number  of  degrees. 

15.  When  it  is  noon  in  Boston,  what  is  the  time  on  a 
line  extending  north  and  south  through  Boston  ? 

16.  This  line   is   called  a  meridian,  or   noonday  line. 


1 

hr. 

15 

4 

^^ 

^^-^^ 

Jflit},  a — TT 

:::^ 

^1 

15 

-> 

4^ 

^1 

^^ 

sec.  .^nT. 

-^ 

16 

ADVANCED   ARITHMETIC.  239 

Show  a  meridian  on  the  globe.    What  is  a  meridian  ?   When 
it  is  noon  in  Boston,  is  it  noon  at  New  York  ?     Explain. 

17.  When  it  is  noon  here,  what  is  the  time  on  the 
meridian  opposite? 

Note.  —  Post  meridian  (p.m.)  means  after  noon,  and  ante  meridian 
(a.m.)  means  before  noon. 

18.  How  many  meridians  are  there  1  degree  apart  ? 

19.  How  many  meridians  15  degrees  apart  ? 

20.  How  many  meridians  are  represented  on  the  globe  ? 

21.  How  many  degrees  apart  are  they? 

22.  What  is  the  shape  of  that  part  of  the  earth's  sur- 
face bounded  by  two  meridians  15  degrees  apart  ? 

23.  Where  is  this  surface  the  widest  ?  Where  are  the 
degrees  the  longest  ?     Why  ? 

24.  Is  the  length  of  a  degree  in  this  latitude  equal  to  a 
degree  on  the  equator  ?     Explain. 

The  length  of  a  degree  on  the  equator  is  60  geographical 
miles.  The  length  of  a  degree  on  the  parallel  60  degrees 
north  of  the  equator  is  30  mi. 

1.  Which  of  two  meridians  moves  under  the  sun's  rays 
first?     Why? 

2.  When  it  is  noon  here,  what  is  the  time  15  degrees 
west  ?     Why  ? 

How  long  will  it  take  the  meridian  15  degrees  west  to 
turn  to  where  we  now  are  ? 

When  the  meridian  15  degrees  west  reaches  here,  where 
will  we  be  ? 

How  much  earlier  is  it  15  degrees  west  than  it  is  here  ? 

3.  How  many  miles  west  on  the  equator  is  it  1  hr. 
earlier  ? 

4.  How  long  does  it  take  the  earth  to  turn  the  900  mi.  ? 

Note.  —  One-sixtieth  of  a  degree  is  called  a  minute  (')  of  longitude. 


240  ADVANCED    ARITHMETIC. 

How  long  to  turn  15  degrees?   30?  45?  20?  25?   16? 
10?  5?  1? 

5.  In  what  part  of  an  hour  does  the  earth  turn  1  de- 
gree ?     In  how  many  minutes  ? 

6.  The  earth  turns  each  of  the  different  numbers  of 
degrees  between  1  and  100  in  what  time? 

7.  How  far  does  the  earth  turn  in  4  min.  ?  in  8?.  in 
12  ?  in  6  ?  in  9  ?  in  15  ?  in  30  ?  in  64  ?  in  1  hr.  ?  in  1  hr. 
8  min.  ?  in  2  hr.  7  min.  ? 

8.  How  many  minutes  of  longitude  in  1  degree  ? 

9.  In  4  min.  of  time  the  earth  turns  how  many  degrees  ? 
How  many  minutes  of  distance  ? 

10.  When  it  is  noon  here,  what  is  the  time  15' east?  15' 
west?  45' east?  45' west?  75' east?  75' west?  85' west? 

11.  In  how  many  seconds  does  the  earth  turn  15'?     In 
what  time  does  it  turn  1'  ? 

Use  the  following  as  a  basis  and  question  one  another  : 
The  earth  turns 

15  degrees  in  1  hr.,  or  60  min. 
1  degree  in  4  min. 
15'  in  1  min.,  or  60  sec. 

1'  in  4  sec. 
15"  in  1  sec. 

12.  When  it  is  noon  here,  what  is  the  time  1'  east? 
1'  west?   3'  east?   3'  west? 

13.  How  far  does  the  earth  turn  in  4  sec.  ?  in  8  sec.  ?  in 
20  sec.  ? 

14.  In  what  time  does  the  earth  turn  37°  16'? 
The  earth  turns  37°  in  2  hr.  28  min. 
The  earth  turns  16'  in  1  min.  4  sec. 

The  earth  turns  37°  16'  in  2  hr.  29  min.  4  sec. 

The  earth  turns  30°  in  how  many  hours  ? 
The  earth  turns  7°  in  how  many  minutes  ? 


ADVANCED    ARITHMETIC. 


241 


The  earth  turns  15'  in  how  many  minutes  ? 

The  earth  turns  1'  in  how  many  seconds  ? 
Then  the  earth  turns  37°  16'  in  what  time  ? 
Show  the  corresponding  time  : 
47°  9';  65°  18';  54°  47';  18°  46'. 

15.  One-sixtieth  of  1'  equals  1"  (1  second  of  longitude). 
How  many  seconds  of  longitude  in  1'  of  longitude  ? 

16.  In  4  sec.  the  earth  turns  how  many  seconds  of  longi- 
tude ?     The  earth  turns  15"  of  longitude  in  what  time  ? 

17.  In   1  sec.  the   earth   turns    how  many  seconds   of 
longitude  ? 

18.  In  what  time  does  the  earth  turn  30"  ?   20"  ?   35"  ? 
17"?  5"?  10"?  46"? 

19.  In  what  time  does  the  earth  turn  37°  46'  20"  ? 
The  earth  turns  37°  in  2  hr.  28  min. 

The  earth  turns  46'  in  3  min.  4  sec. 

The  earth  turns  20"  in  Ij  sec. 

The  earth  turns  37°  46'  20"  in  2  hr.  31  min.  5J  sec. 


Greenwich  Observatory. 


242  ADVANCED   ABITHMETIC. 

20.  How  far  does  the  earth  turn  in  2  hr.  31  min.  5 J 
sec? 

In  2  hr.  the  earth  turns  30°. 
In  31  min.  the  earth  turns  7°  45'. 
In  5i  sec.  the  earth  turns  1'20". 
In  2  hr.  31  min.  5^  sec.  the  earth  turns  37°  46'  20". 

21.  In  what  direction  and  how  fast  would  you  have  to 
travel  on  the  earth's  surface  to  remain  in  the  same  rela- 
tive position  to  the  sun  ?     Explain. 

22.  At  J5  it  is  7  a.m.,  and  at  0  5  a.m.  Which  is  farther 
east  ?     How  many  degrees  ? 

23.  If  your  watch  keeps  Chicago  time,  will  it  indicate 
the  right  time  west  of  Chicago  ?  Why  not  ?  Will  it  be 
fast  or  slow  ?  Are  there  any  directions  in  which  you 
might  travel  from  Chicago  so  that  your  watch  would  indi- 
cate the  right  time  ? 

24.  Locate  Greenwich  on  the  globe. 

25.  What  is  the  greatest  distance  in  degrees  that  any 
place  can  be  from  Greenwich  ?     Why  ? 

26.  Find  on  the  globe  the  meridian  30°  west  of  Green- 
wich ;  another  15°  east  of  Greenwich. 

What  is  the  difference  in  longitude  of  these  two  places  ? 
What  is  their  difference  in  time  ? 

27.  When  it  is  noon  on  the  meridian  15°  east  of  Green- 
wich, what  is  the  time  on  the  meridian  30°  west  of  Green- 
wich ? 

28.  Make  five  questions  similar  to  the  following  :  If  the 
difference  of  time  between  two  places  is  2  hr.  and  30  min., 
what  is  the  difference  in  longitude  ?  Ans.   37°  30'. 

29.  Make  five  questions  similar  to  the  following :  If  a 
place  is  27°  east  of  Cincinnati,  and  another  15°  west,  how 
many  degrees  apart  are  the  two  places  ? 

What  is  the  difference  in  time  ?       Ans.    2  hr.  48  rain. 


ADVAISICED   ARITHMETIC.  243 

Table  of  longitude  from  tlie  meridian  of  Greenwich. : 

Chicago,  87°  37'  45"  west.    Paris,  2°  20'  east. 
Canton,  113°  14'  east.  Philadelphia,  75°  10'  west. 

Edinburgh,  3°  11'  west.     .  Rome,  12°  27'  east. 
London,  5'  48"  west.  San  Francisco,  122°  26'  45"  west. 

Montreal,  73°  34'  west.        Washington,  77°  15"  west. 
New  York,  74°  3'  west. 


Find  the  difference  in  the  sun  time  of  the  following 
cities  : 

1.  Chicago  and  New  York. 

2.  Philadelphia  and  Chicago. 

3.  Montreal  and  New  York. 

4.  Paris  and  Canton. 

5.  Edinburgh  and  Canton. 

6.  Chicago  and  San  Francisco. 

7.  New  York  and  San  Francisco. 

8.  London  and  Rome. 

9.  Rome  and  New  York. 

10.  San  Francisco  and  London. 

What  is  the  difference  in  longitude  corresponding  to  the 
following  diff'erences  in  time  ? 

11.  6  lir.  25  min.  14.   3  hr.  7  min.  11  sec. 

12.  2  hr.  19  min.  15.   2  hr.  40  min.  13  sec. 

13.  11  hr.  45  min.  16.    4  hr.  25  min.  15  sec. 

17.  When  it  is  12  o'clock,  noon,  at  Chicago,  what  is  the 
time  at  Boston  ? 

18.  Do  the  cities  of  the  United  States  keep  the  time  of 
their  own  meridians  ?  How  many  meridians  of  standard 
time  in  the  United  States  ? 

19.  About  how  far  west  of  the  90th  meridian  do  places 
keep  the*  time  of  this  meridian  ?  how  far  east  ? 

20.  How  many  times  would   you  reset  your  watch  in 


^sE^ 


244 


ADVANCED    ARITHMETIC.  245 

traveling  from  Portland,  Me.,  to  San  Francisco  ?     Would 
you  move  the  hands  forward  or  backward  ?  in  returning  ? 

21.  Suppose  you  should  travel  west  around  the  world 
keeping  standard  time,  what  changes  would  need  to  be 
made  in  the  time  of  your  watch  ?  in  traveling  east 
around  the  world  ? 

22.  What  is  the  International  Date  Line  ? 

When  it  is  9  a.m.  Monday,  January  1,  at  London,  is 
it  January  1  all  over  the  world  ?  Where  is  it  not  ?  What 
date  is  it  there  ?  Tell  how  this  is  at  9  p.m.  at  London ;  at 
6  A.M. ;  6  P.M. ;  8  a.m.  ;  8  p.m. 

23.  When  it  is  noon  of  Thursday  at  Chicago,  where  is 
it  Friday  ?     Is  it  Wednesday  anywhere  ? 


1.  I  bought  a  number  of  boxes  of  chalk  for  $2.50,  and 
sold  them  for  $3.75,  gaining  25/ on  each  box.  How  many 
boxes  did  I  buy  ? 

2.  Two  men  start  from  the  same  place  and  travel  in 
opposite  directions,  one  at  the  rate  of  3  mi.  an  hour  and 
the  other  at  -the  rate  of  5  mi.  an  hour.  How  far  apart  are 
they  at  the  end  of  6  hr.  ? 

When  they  are  96  mi.  apart,  how  many  hours  have  they 
traveled  ? 

3.  If  these  men  had  traveled  in  the  same  direction, 
how  far  apart  would  they  have  been  in  6  hr.  ? 

4.  A  silver  dollar  weighs  412.5  grains,  and  contains 
371.25  grains  of  pure  silver.  How  much  of  it  is 
copper  ? 

5.  A  man  having  $b  bought  5  calves  and  had  $y  left. 

^^M^  =  what? 


6.    A  man  having  $m  bought  3  chairs  and  had  $y  left. 
What  equals  the  cost  of  1  chair  ? 


246  ADVANCED    ARITHMETIC. 

7.  Mary  gave  to  each  of  her  playmates  4/  and  had 
20/  left.  If  she  had  given  each  playmate  9/,  it  would 
have  taken  all  her  money.  How  many  playmates  had 
she? 

8.  A  man  bought  land  at  $40  an  acre  and  had  $500 
remaining.  Had  he  paid  $60  an  acre,  it  would  have  taken 
all  his  money.     How  many  acres  did  he  buy  ? 

9.  The  difference  between  two  numbers  is  3050 ;  the 
greater  is  6848.     What  is  the  less  ? 

10.  The  number  of  equal  units  is  6 ;  one  of  the  equal 
units  is  680.     What  is  their  sum  ? 

11.  The  sum  of  6  equal  units  is  4080.  What  is  one  of 
the  equal  units  ? 

12.  A  man  earns  $25.25  as  often  as  a  boy  earns  $9.50, 
and  in  a  certain  time  they  together  earn  $1142.50.  How 
much  does  each  earn  ? 

13.  If  4  men  earn  $8  a  day,  how  many  dollars  will  48 
men  earn  in  1  day  ? 

14.  When  oranges  are  45/  a  dozen,  how  much  must 
you  pay  for  4  ? 

15.  If  3  lemons  cost  10/,  what  do  a  dozen  cost  ? 

16.  What  is  the  cost  of  2  bu.  1  pk.  of  apples  at  40/ 
a  pk.? 

17.  What  does  it  cost  to  grade  40  rd.  of  road  at  $175 
a  mile  ? 

18.  If  18  men  can  build  a  wall  in  1  da.,  what  part  of 
the  wall  can  12  men  build  in  the  same  time  ? 

19.  If  12  men  can  build  f  of  a  wall  in  1  da.,  in  what 
time  can  they  build  the  entire  wall  ? 

20.  If  a  pole  17  ft.  high  casts  a  3-ft.  shadow,  what  is 
the  length  of  a  shadow  cast  by  a  pole  68  ft.  high  ? 

21.  A  pole  150  ft.  high  fell  and  broke  into  two  parts ; 
f  of  the  longer  part  was  equal  to  the  shorter  part. 
What  was  the  length  of  each  part  ? 


ADVANCED    ARITHMETIC.  247 

22.  When  f  of  the  time  past  noon  equals  f  of  the  time 
to  midnight,  what  is  the  time  past  noon  ? 

23.  At  $16f  an  acre,  how  many  6  acres  of  land  can  be 
bought  for  $400  ? 

24.  The  number  of  pupils  in  a  school  is  120  ;  the  number 
of  boys  is  12  more  than  the  number  of  girls.  How  many 
boys  in  the  school  ? 

25.  A  horse  and  carriage  are  worth  $420.  One-half  the 
value  of  the  carriage  equals  ^  the  value  of  the  horse. 
Find  the  value  of  each. 

26.  10  equals  f  of  the  number  of  bushels  of  wheat  a 
man  owns  which  equals  f  of  the  number  of  bushels  of  oats 
he  owns.     How  many  bushels  of  each  has  he  ? 

27.  To  what  scale  is  a  map  drawn  which  represents 
36,000  sq.  mi.  and  is  10  in.  long  and  8  in.  wide  ? 

28.  What  is  the  cost  of  fencing  a  4-by-9  rectangular 
field  of  3J  acres  at  $1.75  a  rod  ? 

29.  A  rectangular  cistern  is  12  ft.  long,  8  ft.  wide,  and 
11  ft.  deep ;  the  water  in  it  is  5  ft.  deep.  If  a  rectangular 
stone  3  ft.  X  2  ft.  X  1  ft.  be  dropped  into  the  cistern,  how 
much  will  the  water  rise  ? 

30.  If  f  of  a  block  on  State  Street  is  worth  $Xj  what 
equals  the  value  of  ^j  of  a  block? 

Express  in  one  sentence  the  answer  to  each  of  the  next 
three  questions  : 

31.  If  y\  of  a  farm  is  worth  $577,  what  equals  the 
value  of  f  of  >the  farm  ?     What  is  the  value  ? 

32.  If  ^1  of  a  lot  is  worth  $850,  what  equals  the  value 
of  the  lot  ?     What  is  the  value  ? 

33.  If  I  of.  a  lb.  of  coffee  costs  26/,  what  does  16i  lb. 
cost? 

34.  If  A  can  do  J-  and  B  -J-  of  a  piece  of  work  in  x  da., 
what  equals  the  time  in  which  they  can  do  the  work 
together  ? 


248  ADVANCED    ARITHMETIC. 

35.  If  5  men  mow  27f  A.  of  grass  in  a  certain  time, 
what  equals  the  number  of  men  that  can  mow  x  A.  in 
the  same  time  ? 

36.  If  3  men  can  mow  20  A.  of  grass  in  8  da.,  how 
many  men  can  mow  10  A.  in  6  da.  ?  Show  this  statement 
to  be  true : 

,.    .       =  number  of  men  required  to  mow  10  A.  in  6  da. 

37.  What  is  the  value  of  a  half  section  of  land  at  $16J 
an  acre  ? 

38.  If  the  expenses  of  a  family  of  12  persons  amount  to 
$80  in  10  wk.,  how  long  will  $400  support  a  family  of 
9  persons  ? 

39.  If  $200  yields  $64  in  4  yr.,  at  the  same  rate  how 
long  does  it  take  $500  to  yield  $160  ? 

What  equals  the  number  of  dollars  $500  would  yield 
in  4  years  ? 

Then  $500  would  yield  $64  in  what  time  ? 

40.  What  will  a  board  20  ft.  long  and  9  in.  wide  cost  at 
$30  a  thousand  ? 

41.  A  girl  sold  flowers  at  20/  and  gained  a  sum  equal 
to  \  of  the  cost.     What  was  the  cost  ? 

42.  If  John  can  build  a  fence  in  |-  of  a  month,  and 
James  in  -J  of  a  month,  in  what  time  can  they  together 
build  it? 

43.  A  woman  bought  246  apples  at  the  rate  of  3  for 
4/,  and  sold  them  at  the  rate  of  2  for  3/.  What  did  she 
gain? 

44.  What  is  the  surface  of  a  brick  2  in.  X  4  in.  X  8  in.  ? 

45.  What  is  the  distance  around  a  room  which  requires 
to  cover  its  floor  5  strips  of  carpet  each  5  yd.  long  and  f  of  \ 
a  yd.  wide  ? 

46.  How  many  half-inch  cubes  in  a  2-in.  cube  ? 
47o    If  a  block  of  wood  1  yd.  in  each  dimension  is  sawed 


ADVANCED    ARITHMETIC.  249 

vertically  through  the  middle  from  front  to  back,  and  ver- 
tically through  the  middle  from  right  to  left,  and  horizon- 
tally through  the  middle,  what  are  the  dimensions  of  one 
of  the  parts  ? 

48.  How  many  16  hundred-thousandths  in  96  thou- 
sandths ? 

49.  Change  yf  ^  to  a  decimal. 

50.  Express  as  a  common  fraction  .00375. 

51.  What  decimal  of  a  ream  is  3.8  quires  ? 

52.  What  decimal  of  an  acre  is  8.6  sq.  rd.  ? 

53.  How  much  water  must  be  added  to  750  gal.  of  wine 
worth  $5  a  gal.  that  the  mixture  may  be  worth  $4  a  gal.  ? 

54.  A  piece  of  paper  is  7  in.  X  11  in.  Cut  it  into  2 
pieces  so  that  there  are  11  sq.  in.  more  in  one  piece  than 
in  the  other. 

55.  Make  and  solve  5  problems  similar  to  the  54th. 

56.  $6820  were  divided  between  two  men.  The  difference 
between  their  shares  was  $512.     What  did  each  receive  ? 

57.  There  were  51,000  votes  cast  at  the  city  election. 
Of  these  one  candidate  received  1550  more  than  the  other. 
How  many  votes  did  each  receive  ? 

58.  a  is  the  perimeter  of  an  equilateral  triangle  and  b 
its  altitude.     What  equals  its  area  ? 

59.  If  the  base  of  a  triangle  is  75  rd.  and  the  altitude 
40  rd.,  how  many  acres  does  it  contain  ? 

60.  At  $375  an  acre,  what  is  the  value  of  a  triangular 
field  whose  base  is  60  rd.  and  altitude  42f  rd.  ? 

61.  At  $.20  a  foot,  what  is  the  cost  of  fencing  a  tri- 
angular field  the  three  sides  of  which  are  in  the  ratio  of  5, 
6,  and  8,  the  shortest  side  being  120  ft.  ? 

62.  The  gable  ends  of  a  barn  are  each  20  ft.  wide,  and 
the  perpendicular  height  of  the  ridge  above  the  eaves  is  8 
ft.  How  many  feet  of  boards  will  be  required  to  boa?d 
both  gables  ? 


250  ADVANCED    ARITHMETIC. 

63.  What  is  the  altitude  of  a  triangle  having  an  area  of 
100  sq.  ft.  and  a  base  of  12^  ft.  ? 

64.  Paid  $16,000  for  a  triangular  farm  at  $800  an  acre. 
If  the  base  of  the  field  is  a  mile,  what  is  its  altitude  ? 

65.  How  many  square  feet  are  there  on  the  sides  and 
bottom  of  a  box  without  cover  2  ft.  X  3  ft.  X  4  f t.  ? 

66.  If  the  oil  in  a  tank  will  run  through  an  opening  3 
in.  X  3  in.  in  20  min.,  in  what  time  will  it  run  through  an 
opening  2  in.  X  2  in.  ? 

67.  At  $5  per  cord,  what  is  a  pile  of  ^ood  worth  that  is 
60  ft.  long,  4  ft.  wide,  and  8  ft.  high  ? 

68.  A  grocer  exchanges  flour  worth  $6  a  barrel  for  wood 
worth  $8  a  cord.  If  the  owner  of  the  wood  asks  $9.50  a 
cord,  to  increase  in  the  same  proportion  what  price  should 
the  grocer  ask  for  his  flour  ? 

69.  Mr.  Jones  invested  $1250  and  gained  an  amount 
equal  to  10%  of  this  sum  every  year  for  3  yr.  What  was 
his  profit  ? 

70.  What  is  the  tax  of  3%  on  property  valued  at  $700  ? 

71.  A  lawyer  received  3%  for  collecting  a  debt  of  $350. 
What  was  his  commission  ? 

72.  What  amount  of  money  will  pay  a  note  of  $200  due 
in2|-yr.  at8%? 

73.  A  paid  $128  for  goods,  which  was  33-J-%  less  than 
their  value.     What  was  their  value  ? 

74.  If  to  30  qt.  of  wine  1  gal.  of  water  is  added,  what 
%  of  the  mixture  is  water  ? 

75.  When  selling  hay,  what  %  does  a  farmer  gain  if 
the  scales  on  which  his  hay  is  weighed  mark  1  cwt.  at  94 
lb.  ?  When  paid  for  18  tons  of  hay  at  $5  a  ton,  how  much 
does  he  gain  ? 

76.  If  by  selling  goods  at  16f  %  profit  a  merchant  clears 
$42,  what  was  the  cost  of  the  goods  ? 

77.  In  a  mixture  of  butter  and  tallow  the  tallow  equals 


ADVANCED   ARITHMETIC.  251 

12^%  of  the  butter.     The  tallow  equals  what  %  of  the 
mixture  ? 

78.  By  selling  a  house  for  $800  a  man  lost  10%.  What 
should  he  have  sold  it  for  to  gain  10%  ? 

79.  A  boy  paid  $1.50  for  60  newspapers.  Allowing 
that  J  will  be  unsold,  for  how  much  apiece  must  he  sell 
the  others  to  gain  33-J%  ? 

80.  A  2-ft.  square  equals  how  many  4ths  of  a  square 
foot? 

81.  27  equals  how  many  4ths  of  12  ? 

82.  9  ft.  equals  how  many  8ths  of  12  ft.  ? 

83.  4  ft.  equals  how  many  7ths  of  3  ft.  ? 

84.  One  of  the  faces  of  a  -J-in.  cube  equals  what  %  of 
one  of  the  faces  of  a  1-in.  cube  ? 

85.  E  bought  a  basket  of  grapes  for  f  of  the  market 
price.     The  saving  equaled  what  part  of  the  cost? 

86.  If  $3  is  lost  by  selling  5  yd.  of  velvet  at  a  loss  of 
30%,  what  was  the  cost  of  the  velvet  per  yd.? 

87.  A  milliner  bought  $150  worth  of  goods  and  sold 
them  for  $120.  How  many  cents  did  she  lose  on  each 
dollar  invested  ? 

88.  If  a  merchant  sells  f  of  his  stock  for  a  sum  equal  to 
the  cost  of  f  of  it,  what  %  is  gained  ? 

89.  What  is  the  thickness  of  a  block  of  stone  which 
weighs  1650  lb.  and  is  3  ft.  long  and  2  ft.  wide,  if  a  cubic 
foot  of  stone  weighs  150  lb.  ? 

90.  A  room  30  ft.  long  requires  80  yd.  of  carpet  f  of  a 
yd.  wide.     How  wide  is  the  room  ? 

91.  A  brick  is  8  in.  X  4  in.  X  2  in.  How  many  will  it 
take  to  lay  ^  mi.  of  8-ft.  sidewalk,  the  brick  being  laid  on 
one  of  the  largest  surfaces  ? 

92.  I  send  my  agent  $420  to  invest  in  plows  at  $10 
each  after  deducting  his  commission  at  5%.  How  many 
plows  can  be  purchased? 


252  ADVANCED    ARITHMETIC. 

93.  What  is  the  tax  collected  on  $6000  worth  of  prop- 
erty at -J  %? 

94.  When  the  rate  of  taxation  is  J%,  what  amount  of 
property  will  give  a  tax  of  $18.75? 

95.  At  what  rate  must  property  valued  at  $100,000  be 
taxed  to  raise  $500  ? 

96.  How  many  board  feet  of  lumber  are  required  to 
make  from  2-in.  lumber  a  cubical  box  that  will  hold  1  cu. 
ft.  water  ? 

97.  How  many  pickets  will  be  required  to  fence  a  gar- 
den 8  ft.  X  12  ft.  if  the  pickets  are  4  in.  wide  and  are 
placed  2  in.  apart  ? 

98.  If  a  horse  eats  2  bu.  of  oats  in  x  da.,  in  how  many 
X  da.  will  2  horses  eat  26  bu.  ? 

99.  If  4  men  plow  x  A.  in  8^  hr.,  how  many  x  A.  can  8 
men  plow  in  20  hr.  ? 

100.  If  f  of  a  yd.  of  ribbon  costs  $f,  what  does  \  yd. 
cost? 

101.  Show  by  diagram  the  answer  to  the  following :  f  of 
a  unit  increased  by  J  of  the  unit  and  this  increased  by  the 
unit  34  will  give  twice  the  unit.     What  is  it  ? 

102.  A  man  bought  peaches  at  $1  a  pk.,  and  half  as 
many  pecks  of  apples  at  half  as  much  a  peck.  The  cost 
of  both  was  $12.50.  How  many  pecks  of  each  did  he 
buy? 

103.  A  merchant  bought  a  number  of  yards  of  silk  at 
$3  a  yd.,  and  \  as  many  yards  of  cashmere  at  \  as  much 
a  yd.  The  cost  of  the  whole  was  $175.  How  many  yards 
of  each  did  he  buy  ? 

104.  A  man  sold  f  of  his  corn  crop  and  had  720  bu.  left. 
How  much  had  he  at  first  ? 

105.  The  sum  of  two  numbers  is  17935,  and  their  differ- 
ence equals  f  of  the  greater  number.  What  are  the  num- 
bers ? 


ADVANCED    ARITHMETIC.  253 

106.  f  of  a  bbl.  of  apples  at  $7  a  bbl.  will  pay  for  what 
part  of  a  ton  of  coal  at  $9  a  ton  ? 

107.  If  goods  are  bought  at  f  off  the  retail  price,  what 
%  is  gained  by  selling  at  the  retail  price  ? 

108.  An  agent  collected  50%  of  a  debt  of  $1200  and 
charged  3^%  commission.     What  amount  will  he  remit  ? 

109.  By  buying  pears  at  the  rate  of  3  for  6/,  ^  is  saved. 
What  is  the  retail  price  per  pear  ? 

110.  At  what  price  must  cloth  costing  $1.20  a  yd.  be 
marked  so  that  10%  may  be  deducted  from  the  marked 
price  and  it  be  sold  at  a  profit  of  20%? 

111.  Find  the  difference  between  37^%  of  $2400  and 
.37^%  of  the  same. 

112.  What  %  of  an  A.  is  30  sq.  rd.  ?  5  sq.  ft.? 

113.  i  a  mi.  is  what  %  of  40  rd.  ? 

114.  A  merchant's  assets  are  $27,000  and  his  liabilities 
$87,000.     What  %  can  he  pay  ? 

115.  The  premium  for  insuring  a  stock  of  goods  at  1^% 
was  $250.     What  was  the  amount  invested  ? 

116.  If  60  apples  are  worth  $1  in  currency,  what  in 
currency  are  100  apples  worth  ? 

117.  When  a  paper  dollar  is  worth  60/  in  gold,  what  is 
the  price  of  gold  ? 

118.  Property  was  bought  for  $2759  and  sold  for  $2516. 
What  was  the  %  of  loss  ? 

119.  Sold  land  at  120%  of  its  cost  and  thereby  gained 
$7  a  ft.     What  was  the  cost  per  ft.  ? 

120.  By  selling  a  house  for  $1200  I  shall  lose  12^%. 
For  how  much  should  I  sell  it  to  gain  12-^%  ? 

121.  A  fruit  dealer  lost  33^%  of  a  lot  of  apples  and 
sold  the  remainder  at  a  gain  of  50%.  Eequired  the  %  of 
gain  or  loss. 

122.  A  farmer  lost  16|%  of  his  stock  and  sold  the  re- 
mainder at  a  profit  of  50  % .  What  was  the  %  of  gain  or  loss  ? 


264  ADVANCED   ARITHMETIC. 

123.  A  boy  bought  a  knife  for  60/  and  sold  it  for  20% 
more  than  it  cost  and  25%  less  than  he  asked  for  it.  How 
much  did  he  ask  ? 

124.  A  farmer  sold  a  cow  to  a  shipper  at  a  gain  of  20  % ; 
the  shipper  sold  to  a  butcher  at  a  gain  of  33-J-% ;  the  butcher 
realizing  $60  for  the  cow  made  10%.  What  was  the  price 
the  farmer  received  for  the  cow  ? 

125.  What  is  the  entire  surface  of  a  block  of  wood  1^ 
ft.  X  5i  ft.  X  3  f t.  ? 

126.  How  many  cords  of  wood  can  be  piled  against  the 
wall  of  a  room  16  ft.  X  16  ft.  X  8  ft.,  so  that  one  end  of 
each  stick  will  touch  the  wall  ? 

127.  If  berries  are  bought  at  the  rate  of  12  qt.  for  $1 
and  sold  at  8  qt.  for  $1,  what  is  the  %  of  profit  ? 

128.  A  man  invests  his  money  in  wheat,  which  sells  for 
$2640  more  than  he  paid  for  it.  His  money  then  equals 
133-J%  of  what  he  had  at  first.  How  much  did  he  have  at 
first? 

129.  By  selling  wine  at  $1.80  a  pt.  a  dealer  gains  40%. 
What  would  be  the  selling  price  at  a  loss  of  40%  ? 

130.  A  merchant  sold  lead  pencils  for  3/  each,  which 
was  at  a  gain  of  Ih^o-     What  did  the  pencils  cost  him? 

131.  At  what  rate  %  simple  interest  will  $100  double 
itself  in  10  yr.  ? 

132.  If  by  buying  a  5-lb.  box  of  paper  for  $1,  J  is  saved, 
what  is  the  cost  when  bought  by  the  single  pound  ? 

133.  A  strip  of  land  containing  2  A.  is  :J^  mi.  long.  How 
wide  is  it  ? 

134.  A  and  B  can  mow  a  field  in  4  da.,  and  A  can  do 
only  I  as  much  work  as  B.  How  long  will  it  take  each 
alone  to  mow  the  field  ? 

135.  If  a  creditor  agrees  to  accept  60/  on  the  dollar 
and  then  discounts  3%  for  immediate  payment,  what  does 
he  lose  on  a  claim  of  $28,000  ? 


ADVANCED   ARITHMETIC.  255 

136.  At  what  discount  must  stock  paying  a  half-yearly- 
dividend  of  3|-%  be  purchased  to  enable  the  buyer  to  real- 
ize 8%  on  his  investment  ? 

137.  How  high  must  a  box  be  to  hold  80  2-in.  cubes  if 
its  other  dimensions  are  8  in.  ?     Why  ? 

138.  What  is  the  area  of  the  entire  surface  of  a  4-ft.  cube  ? 

139.  How  wide  must  a  paper  border  of  a  square  yard  be 
to  contain  432  sq.  in.  ? 

140.  How  many  paper  boxes  3  in.  long,  2  in.  wide,  and 
2  in.  deep  can  be  packed  in  a  cu.  ft.  of  space  ? 

141.  A  sidewalk  a  mile  long  contains  f  of  an  acre. 
How  wide  is  it  ? 

142.  If  your  schoolroom  is  16  ft.  high  and  contains 
12,000  cu.  ft.,  how  many  sq.  ft.  are  there  in  the  floor  ? 

143.  If  a  cu.  ft.  of  water  weighs  1000  oz.  and  expands 
j\  in.  in  freezing,  how  many  tons  of  ice  are  there  on  a  4- A. 
pond  when  the  pond  is  frozen  8  in.  deep  ? 

144.  What  is  the  cost  of  painting  a  blackboard  20  ft. 
long  and  3^  ft.  wide  at  36/  a  sq.  yd.  ? 

145.  How  many  cu.  ft.  in  5^  cd.  of  wood  ? 

146.  A  reaper  cuts  a  6-ft.  swath  around  a  field  that  is 
48  rd.  long  and  540  ft.  wide.  How  many  sq.  ft.  in  the 
swath  ? 

147.  How  high  do  you  ascend  in  going  up  3  flights  of 
stairs,  the  first  of  15  steps,  and  the  other  two  14  steps 
each,  if  each  step  is  9^  in.  high  ? 

148.  What  is  the  cost  of  5  boards,  each  f  in.  thick,  16 
ft.  long,  and  18  in.  wide,  at  $30  per  M.  ? 

149.  How  many  rods  of  fence  will  be  required  to  enclose 
a  rectangular  field  containing  20  A.,  one  side  of  which  is 
128  rd.? 

150.  A  reservoir  is  36  ft.  long  and  25  ft.  3  in.  wide. 
How  many  cu.  ft.  of  water  must  be  drawn  off  to  lower  the 
surface  4  in.  ? 


256  ADVANCED   AKITHMETIC. 

151.  How  many  posts  6  ft.  apart  will  enclose  a  section 
of  land  ? 

152.  If  a  map  is  drawn  to  the  scale  of  125  mi.  to  an  in., 
what  is  the  area  of  a  state  the  dimensions  of  which  on  the 
map  are  If  in.  by  2  in.  ? 

153.  If  1000  shingles  laid  4  in.  to  the  weather  will  cover 
100  sq.  ft.  of  surface,  how  many  shingles  so  laid  will  be 
required  for  the  roof  of  a  barn  each  side  of  which  is  40  ft. 
wide  and  120  ft.  long  ? 

154.  A  rectangle  3^  in.  long  and  J  in.  wide  equals  what 
part  of  a  2-in.  square  ? 

155.  How  many  feet  of  lumber  in  a  stick  of  timber  30 
ft.  long  and  25  in.  square  ? 

156.  John  and  William  can  saw  a  pile  of  wood  in  14  da., 
and  John  can  saw  only  f  as  much  as  William.  In  what 
time  can  each  alone  saw  the  wood  ? 

157.  If  X  is  the  number  of  men  required  to  do  a  piece 
of  work  in  a  given  time,  how  many  men  will  be  required 
to  do  4  times  as  much  work  in  -J-  of  the  time  ? 

158.  If  8/  buys  a  6-oz.  loaf  when  flour  is  $10  a  bbl., 
how  large  a  loaf  does  it  buy  when  flour  is  $7  a  bbl.  ? 

159.  If  ^^  is  saved  by  buying  a  dozen  handkerchiefs 
for  $4.50,  what  is  the  price  of  a  handkerchief  if  bought 
singly  ? 

160.  If  7  yd.  of  goods  are  required  for  a  suit  and  the 
cloth  shrinks  10%  in  sponging,  how  many  yards  must  I 
buy  for  a  suit  ? 

161.  If  13  yd.  of  gingham  are  required  to  make  a  dress, 
which  shrinks  ^^  in  washing,  how  many  yards  must  be 
bought  to  make  a  dress,  allowing  for  shrinkage  ? 

162.  If  canvas  shrinks  10%,  how  many  yards  should  be 
purchased  so  that  after  shrinking  it  may  cover  a  platform 
27  ft.  long  and  26  ft.  wide  ?  a  platform  7  yd.  long  and  h\ 
yd.  wide? 


ADVANCED    ARITHMETIC.  257 

163.  If  you  buy  wool  at  24/  a  lb.  and  it  loses  |  of  its 
weight  in  cleansing,  for  how  much  must  it  be  sold  to  make 
10/  per  lb.  ? 

What  remains  of  a  lb.  after  cleansing? 

For  what,  then,  must  f  of  a  lb.  be  sold 
— ^^ — =  ?     that  there  may  be  no  loss  ? 

For  what,  then,  must  f  of  a  lb.  be  sold 
that  10/  may  be  gained  ?  For  what,  then,  must  a  lb.  be 
sold? 

164.  The  sum  of  A  and  B  equals  x.  f  of  ^  equals  f  of 
B.     A  equals  what  part  of  ic  ? 

What  is  the  relation  of  ^  to  f  of  ^  ? 
What,  then,  is  the  relation  of  A  to  f  oi  B? 
I  of  f  equals  what  ? 
What,  then,  is  the  relation  of  ^  to  ic  ? 

165.  The  sum  of  C  and  D  equals  Y.  f  of  (7  equals  f  of 
I).     What  is  the  ratio  of  C  to  Y? 

166.  The  time  past  noon  and  the  time  to  midnight  equals 
what  ?  f  of  the  time  past  noon  equals  f  of  the  time  to 
midnight.     What  is  the  time  ? 

What  is  the  relation  of  the  time  past  noon  to  f  of  the 
time  past  noon  ? 

What,  then,  is  the  relation  of  the  time  past  noon  to  f  of 
the  time  to  midnight  ? 
-— -  =  ?         I  of  f  equals  what  ? 

What,  then,  is  the  relation  of  the  time  past 
noon  to  the  time  to  midnight  ? 

What,  then,  is  the  relation  of  the  time  past  noon  to  the 
time  from  noon  until  midnight  ? 

167.  B  is. a?  mi.  in  advance  of  A.  A  travels  3  times  as 
fast  as  B.  What  equals  the  distance  A  travels  before 
overtaking  B  ? 

The  distance  B  travels  in  any  time  equals  what  part  of 
the  distance  A  travels  in  the  same  time  ? 


258  ADVANCED   ARITHMETIC. 


Then  tha  distance  B  is  in  advance  of  A  equals 
— —  =  ?     what  part  of  the  distance  A  travels  ?     Why  ? 

What  is  the  relation  of  the  distance  that  A 
travels  to  the  distance  B  is  in  advance  of  A  at  the  time  of 
starting  ? 


168.  What  ratios  do  you  find  in  each  of  these  diagrams  ? 
Make  problems  of  the  nature  of  those  above  and  involving 
these  ratios. 

169.  A  car  travels  8  times  as  fast  as  a  tramp.  The 
tramp  is  x  mi.  in  advance  of  the  car.  What  equals  the 
distance  the  car  must  travel  before  overtaking  the  tramp  ? 

In  any  time,  the  distance  the  tranip  travels  equals  what 
part  of  the  distance  traveled  by  the  car  ? 
„  .  The  distance  the  tramp  is  in  advance  of  the  car 

—=~  —  ?     equals  what  part  of  the  distance  the  car  travels  ? 
What   is   the   ratio  of  the    distance  the  car 
travels  to  the  distance  the  tramp  is  in  advance  of  the  car 
at  the  time  of  starting  ? 

170.  The  minute-hand  moves  how  many  times  as  fast  as 
the  hour-hand  ?  In  any  time  the  distance  moved  by  the 
hour-hand  equals  what  part  of  the  distance  moved  by 
the  minute-hand  ?  The  hour-hand  is  x  minutes  in  advance 
of  the  minute-hand.     What  equals  the  time  in  which  the 


ADVANCED    ARITHMETIC.  269 

minute-hand  will  overtake  the  hour-hand  ?  What  is  the 
ratio  of  the  distance  the  hour-hand  is  in  advance  of  the 
minute-hand  to  the  distance  the  minute-hand  moves  before 
overtaking  the  hour-hand  ? 

What  is  the  ratio  of  the  distance  moved  by  the  minute- 
hand  to  the  distance  the  hour-hand  is  in  advance  of  the 
minute-hand  at  the  time  of  starting? 

11 

171.  What  is  the  time  of  day  when  f  of  the  time  past 
midnight  equals  |  of  the  time  to  noon  ? 

172.  What  is  the  hour  of  day  when  f  of  the  time  past 
noon  equals  f  of  the  time  to  midnight  ? 

173.  If  the  time  past  noon  increased  by  40  min.  equals 
I  of  the  time  from  noon  to  midnight,  what  time  is  it  ? 

174.  At  what  time  between  5  and  6  o'clock  are  the  hour 
and  minute  hands  of  a  watch  together  ?  At  what  time 
between  3  and  4  o'clock  ? 

175.  A  runs  12  rd.  while  B  runs  11.  In  a  race  of  6  min., 
how  much  must  B  be  given  the  start  that  they  may  reach 
the  goal  together  ? 

176.  John  is  30  rd.  in  advance  of  Frank,  but  Frank  runs 
8  yd.  while  John  runs  5.  How  far  must  Frank  run  to 
overtake  John  ? 

177.  If  a  steamer  sails  8  mi.  an  hr.  down  stream  and  6 
mi.  an  hr.  up  stream,  how  far  can  it  go  down  and  back  in 
12  hr.? 

178.  Alice  starts  to  school,  and  after  she  has  traveled  ^ 
of  the  distance  George  starts  from  the  same  place  to  over- 
take her,  traveling  three  times  as  fast.  What  part  of  the 
distance  from  the  starting  point  to  the  schoolhouse  does 
George  travel  before  overtaking  Alice  ? 

179.  Make  and  solve  3  problems  similar  to  the 
178th. 


260  ADVANCED    ARITHMETIC. 

180.  How  many  miles  does  a  boy  walk  in  plowing  a 
12- A.    square  field  and  turning  a  10-in.  furrow  ? 

181.  Stock  bought  at  12%  premium  pays  5%  on  the  in- 
vestment.    'What  would  it  pay  if  bought  at  15%  discount  ? 

182.  When  stock  is  bought  at  a  discount  of  20%  and 
sold  at  a  discount  of  8%,  what  %  is  gained  ? 

183.  What  was  the  face  of  a  draft  bought  for  $2730, 
when  exchange  was  at  ^%  discount  ? 

184.  If  stock  bought  at  10%  discount  pays  8%  on  the 
investment,  at  what  price  should  the  same  stock  be  bought 
to  pay  10%? 

185.  If  gold  is  at  16f  %  premium,  what  is  the  corre- 
sponding discount  on  currency  ? 

186.  If  a  man  asks  40%  profit,  but  falls  10%  on  his 
asking  price,  what  is  the  rate  of  profit  ? 

187.  A  man  sold  a  cow  for  $45  and  thereby  gained  2^^0' 
If  he  had  sold  it  for  $30,  would  he  have  gained  or  lost, 
and  what  %  ? 

188.  A  grocer  invested  $215  as  follows : 

He  spent  60%  of  the  money  for  pears  at  $1.50  a  bu.; 
30%  for  apples  at  87^/  a  bu.,  and  the  remainder  for  pota- 
toes at  $.40  a  bu.     How  many  bu.  of  each  did  he  buy  ? 

189.  Two  horses  were  sold  for  $189  each.  On  one  there 
was  a  gain  of  10%  and  on  the  other  a  loss  of  10%.  Find 
the  gain  or  loss  on  the  sale  of  both. 

190.  A  man  has  two  pictures  and  a  frame  worth  $20. 
If  he  puts  the  frame  on  the  first  picture,  it  will  be  worth 
f  as  much  as  the  second  picture ;  but  if  he  puts  the  frame 
on  the  second  picture,  it  will  be  worth  2f  times  as  much  as 
the  first  picture.     What  is  the  value  of  the. pictures  ? 

191.  My  semi-annual  income  from  a  stock  investment 
that  yields  10%  is  $106.     How  many  shares  do  I  own? 

192.  A  lawyer  earns  $450  collecting  money  at  5%. 
What  amount  does  he  collect  ? 


ADVANCED    ARITHMETIC.  261 

193.  If  goods  are  bought  at  J  of  their  value  and  sold 
for  25%  more  than  their  value,  what  is  the  gain  %  ? 

194.  A  florist  buys  flowers  at  -J-  off  from  the  retail  price. 
What  %  does  he  make  by  selling  at  the  retail  price  ? 

195.  If  wool  that  costs  $.60  per  lb.  shrinks  5%  in 
cleansing,  at  what  price  per  lb.  must  it  be  sold  to  gain 
33^%  on  the  cost? 

196.  I  buy  8%  stock  at  110.  What  %  do  I  receive  on 
my  investment  ? 

197.  A  merchant  received  for  a  lot  of  goods  $6520.  He 
had  deducted  5%  from  the  face  of  the  bill  and  still  found 
that  he  had  made  12%.     What  did  he  pay  for  the  goods  ? 

198.  If  $80  yields  $6.30  in  a  given  time,  what  principal 
will  yield  $104.50  in  the  same  time  ? 

199.  If  $475  yields  $x  in  a  certain  time,  what  will 
$187.50  yield  in  the  same  time  ? 

200.  A  road  f  mi.  long  and  5  rd.  wide  is  run  through  a 
man's  farm.  He  is  paid  $30  an  acre  for  his  land.  How 
much  does  he  receive  ? 

201.  f  of  Mr.  Wilson's  property  equals  f  of  Mr.  Brown's. 
The  difference  in  the  value  of  the  two  men's  property  is 
$724.     What  is  the  value  of  each  man's  property  ? 

202.  The  fore  wheels  of  a  carriage  are  each  9  ft.  in  cir- 
cumference, and  the  hind  wheels  are  each  1^  times  as  many 
ft.  in  circumference.  If  each  fore  wheel  revolve  6300 
times  in  going  a  certain  distance,  how  many  times  will 
each  hind  wheel  revolve  ? 


^^' 


SPEER'S    ARITHMETICS 

By  WILLIAM    W.    SPEER, 

Assistant  Superintendent  of  Schools y  Chicago,  III. 

A  Primary  Arithmetic.  Part  I.,  First  Year  Work.  For  teachers. 
i2mo.  Cloth.  Illustrated.  154  pages.  To  teachers  and  for 
introduction,  35  cents. 

An  Elementary  Arithmetic.     Part  II. :   2d,  3d,  and  4th  Years. 

For  pupils.    i2mo.    Cloth.     Illustrated.     314  pages.     For  intro- 
duction, 45  cents. 

An  Advanced  Arithmetic.     Part  III. :  Grammar  Grades. 

[/«  preparation. 
NUMBER  BLOCKS.     To  accompany  Speer's  Arithmetics. 

Standard  Set $2. so- 
Decimal  Set  (to  supplement  Standard  Set) „ 2.50 

Special  Set  (Spheres) 30 

Speer's  Arithmetics  advance  the  study  from  the  science 
of  number  to  that  of  definite  relations  of  quantity. 

They  break  down  the  artificial  barriers  built  between 
groups  of  problems  that  are  alike  in  process  and  differ 
only  in  trade  usage. 

They  recognize  sense-training  as  the  basis  of  thought, 
and  definite  relations  of  magnitudes  as  the  only  basis  of 
mathematical  inferences. 

They  make  simple  ratios  the  key  to  the  solution  of  all 
problems,  and  thus  unify  all  parts  of  the  work  and  save 
much  time. 

They  unfold  the  pupil's  mind,  not  only  by  leading  him 
to  perceive,  but  also  to  form  judgments  and  to  reason. 

The  philosophy  which  underlies  Speer's  Arithmetics  is 
correct,  and  has  won  the  marked  approval  of  advanced 
educators.  It  is  a  long  stride  in  the  right  direction  of  a 
complete  reformation  in  the  teaching  of  number. 


GINN   &  COMPANY,  Publishers, 

Boston.  New  York.  Chicago.  Atlanta.  Dallas. 


FRYE'S  GEOGRAPHIES 


FRYE'S  ELEMENTS  OF  GEOGRAPHY.  Small  4to.  Cloth. 
164  pages.     Fully  illustrated.     For  introduction,  65  cents. 

FRYE'S  COMPLETE  GEOGRAPHY.  With  an  Appendix  con- 
taining 24  pages  of  Reference  Maps.  Large  4to.  Cloth.  184 
pages.     Profusely  illustrated.     For  introduction,  $1.25. 

Frye's  geographies  are  universally  recognized  as  rep- 
resenting an  advance  movement  in  education,  but^  their 
crowning  merit  lies  in  the  fact  that  they  not  only  can 
secure  better  results  than  other  books,  but  can  make  the 
pupils'  work  more  interesting  and  the  teacher's  work 
lighter. 

The  success  of  Frye's  geographies,  which  is  literally 
unparalleled  in  the  history  of  text-book  publishing,  shows 
that  there  is  a  deep  and  widespread  demand  for  the  best 
ideas,  methods,  and  books.  It  goes  without  saying,  that, 
at  the  beginning,  it  requires  a  little  more  skill  and  pains 
to  teach  the  pupils  to  get  ideas  instead  of  words  from 
their  text-books ;  but  that  is  of  course  the  only  right  way, 
and  in  a  little  time  it  becomes  the  easier  and  more 
pleasant. 

L.  H.  Jones,  Superintendent  of  Public  Schools^  Cleveland,  Ohio: 

I  am  delighted  with  Frye's  Complete  Geography ;  it  is  thoroughly 
modern  and  in  line  with  the  best  methods  of  teaching. 

F.  F.  Murdock,  Principal  of  State  Normal  School,  North  Adams,  Mass. 
I  make  no  reservation  when  I  say  that  Frye's  Complete  is  the  best 
grammar-school  geography  I  have  ever  used. 

J.  R.  Miller,  Superintendent  of  Schools,  Big  Rapids,  Mich. 

I  have  examined  a  number  of  the  best  geographies  published  and 
after  a  very  careful  comparison  I  have  recommended  Frye's  geographies 
(Elements  and  Complete)  as  the  best  on  the  market. 


GINN  &  COMPANY,  Publishers, 

Boston.  New  York.  Chicago.  Atlanta.  Dallas. 


MONTGOMERY'S  AMERICAN  HISTORY 

By  D.  H.  MONTGOMERY, 

Author  of  the  "  Leading  Facts  of  History  SeriesP 


Half  leather.    367  pages.  With  full  maps,  full-page  illustrations, 
appendices,  etc.    For  introduction,  $1.00. 


The  author  of  this  book  is  Mr.  D.  H.  Montgomery,  the  eminent 
and  successful  writer  of  historical  text-books. 

Its  general  plan  and  style  will  be  found  similar  to  the  other 
histories  by  this  author.  The  main  difference  is  that  the  American 
History  is  adapted  for  younger  pupils. 

The  greatest  merit  of  the  book  is  in  the  judgment  with  which 
the  leading  events  in  the  development  of  our  country  have  been 
selected  and  the  vividness  with  which  they  are  placed  before  the 
reader's  mind. 

It  has  been  written  and  not  merely  compiled.  Hence  it  has 
an  interest  and  charm  like  that  of  a  story  told  by  an  eyewitness 
of  the  events. 

Not  only  are  the  important  events  clearly  and  luminously 
sketched,  but  their  causes  are  fully  traced  and  the  results  of  all 
important  events  adequately  shown. 

The  author  has  treated  all  subjects  impartially,  following  the 
course  of  events  as  an  eyewitness  elevated  above  the  plane  of 
contention. 

Accuracy  has  been  diligently  and  patiently  studied,  and  investi- 
gations of  original  documents  have  been  made  where  leading 
authorities  have  been  found  to  disagree. 

Every  section  ends  with  a  brief  summary.  Copious  notes  are 
added,  with  many  cross-references.  The  book  contains  an  un- 
usually large  number  of  maps  besides  numerous  fine  engravings 
carefully  selected  as  historical  illustrations.  Chronological  and 
statistical  tables,  a  list  of  reference  books,  index,  questions,  the 
Declaration  of  Independence,  and  the  Constitution  have  been 
added. 

Montgomery's  Leading  Facts  of  American  History  is  full  of 
thought  and  intellectual  life.  It  has  been  pronounced  by  the 
educational  jury  the  best  text-book  on  the  subject,  —  accurate, 
philosophical,  unprejudiced,  of  rare  interest,  easily  handled  by 
teachers,  and  easily  grasped  by  children. 


GINN   &  COMPANY,  Publishers, 

Ppstpn*  New  York.  Chicago.  Atlanta.  Dallas. 


"UNIQUE,  PRACTICAL,  AND  FULL  OF  COMMON  SENSE." 


PRINCE'S  ARITHMETIC  BY  GRADES 

By  JOHN  T.  PRINCE, 

Agent  of  the  Massachusetts  Board  of  Education^  and  author  of '■^Courses  of 
Studies  and  Methods  of  Teaching.'''' 

FOR  INDUCTIVE  TEACHING,  DRILLING,  AND  TESTING. 

EIGHT  BOOKS.    Square  i2mo.    Flexible  cloth.    From  86  to  1 18  pages 
each.    For  introduction,  20  cents. 

TEACHERS'  MANUAL.     For  Teachers  using  Arithmetic  by  Grades. 
Square  i2mo.     Cloth.     225  pages.     For  introduction,  80  cents. 

This  unique  and  attractive  series  of  text-books  consists  of  a  Teachers' 
Manual  and  eight  small  books  for  pupils,  arranged  somewhat  on  the 
lines  of  classification  in  city  graded  schools. 

The  idea  of  a  neat,  compact,  and  inexpensive  drill  book  for  each 
grade,  with  a  manual  for  the  teachers,  is  showing  results  in  a  larger 
amount  of  work  accomplished,  a  more  thorough  understanding  of  prin- 
ciples, and  greater  ease  in  applying  them  ;  also  in  convenience,  neat- 
ness, and  economy  of  wear. 

Many  good  teachers  assure  us  that  the  Prince  books  bid  fair  to  drive 
out  eventually  the  old-fashioned  text-book  in  arithmetic  from  the  field. 

The  Teacher's  Manual  is  devoted  to  suggestions  as  to  methods  of 
teaching  and  drilling,  as  well  as  the  illustrative  processes,  explanations, 
rules,  and  definitions  which  belong  to  the  teacher  to  develop  analytically. 


L.  H.  JONES,  5upt.  of  Schools, 
Indianapolis,  Ind. :  The  plan  is  unique, 
and  withal  so  practical  and  full  of  common 
sense  that  one  does  not  well  see  why  it 
had  not  been  followed  by  others  before. 
The  books  seem  admirably  adapted  to 
their  grades,  and  supply  excellent  work 
for  practice.  In  doing  this  work,  Mr. 
Prince  has  again  laid  the  educational 
world  under  obligations. 

EDWARD  SEARING,  Pres.  of  State 
Normal  School,  Mankato,  Minn.:  After 
a  full  inspection  and  comparison  of  several 
different  series,  we  have  finally  decided  to 
select  your  Prince's  Arithmetics,  as  best 
suited  to  the  needs  of  the  work  in  our 
grades. 

Supt.  F.  TREUDLEY,  Youngs= 
town,  Ohio  :  I  think  it  a  most  admirable 
series  of  books,  full  of  suggestive  prob- 
lems, and  of  a  character  to  insure  mental 
development. 


GEORGE  H.  MARTIN,  Supervisor 
of  Schools,  Boston :  It  is  not  only  unique 
in  place  among  American  arithmetics,  but 
it  abounds  in  novel  features,  which  are 
seen  at  once  to  be  both  philosophical  and 
practical. 

ELLEN  HYDE,  Prin.  of  Normal 
School,  Framingham,  Mass.  :  These 
little  books  take  a  much  needed  and  long 
step  in  the  right  direction,  —  the  direc- 
tion of  thoroughness  and  self-dependence. 
Their  careful  grading,  the  large  amount 
of  drill  and  constant  reviews  tend  to 
thoroughness. 

M.  L.  PALMER,  Supt.  of  Schools, 
Jackson,  Mich. :  Prince's  Arithmetics 
are  giving  us  excellent  satisfaction  in  our 
first  five  grades.  The  books  are  logical, 
are  well  arranged  as  to  subject  matter, 
and  are  intensely  practical.  I  can  recom- 
mend them  as  the  best  I  have  ever  seen. 


GINN  &  COMPANY,  Publishers,  Boston,  New  York,  Chicago. 


MATHEMATICAL  TEXT-BOOKS. 

For  Lower  Grades. 

Bacon:  Four  Years  in  Number $0.40 

Saldwin :         Industrial  Primary  Arithmetic 46 

Gay :  Business  Book-keeping : 

Single  Entry 66 

Double  Entry 1.12 

Complete  Edition 1.40 

Oinn  and  Coady :  Combined  Number  and  Language  Lessons : 

Teachers'  Manual 50 

Tablets  for  Seat  Work.    Blocks  I,  II,  III,  IV Each        .08 

Page :  Fractions : 

Teachers'  Manual 30 

Pupils'  Edition 30 

Prince :  Arithmetic  by  Grades : 

Books  I  to  VIII Each        .20 

Teachers'  Manual : 

Teachers'  Manual,  Part  1 35 

Teachers'  Manual,  Part  II 50 

Complete  Edition 80 

Shove:  Primary  Number  Cards 26 

Speer :  Arithmetics : 

Parti.    Primary.    For  teachers 36 

Part  II.    Elementary.    For  pupils 46 

Part  III.    Advanced 

Wentworth:  Elementary  Arithmetic 30 

Practical  Arithmetic 66 

Mental  Arithmetic 30 

Primary  Arithmetic 30 

Grammar  School  Arithmetic 66 

First  Steps  in  Algebra 60 

Wentworth  and  Hill :  Exercises  in  Arithmetic : 

Exercise  Manual 50 

Examination  Manual 35 

Complete  in  one  volume 80 

Wentworth  and  Eeed :  First  Steps  in  Number : 

Pupils'  Edition 30 

Teachers' Edition 90 

Part  L    First  Year 30 

Part  IL    Second  Year 30 

Partm.    Third  Year 30 


Descriptive  Circulars  of  the  above  books  sent,  postpaid,  on  application. 


GINN    &    COMPANY,    Publishers, 

Boston.  New  York.  Chicago.  Atlanta.  Dallas. 


^wfoD^s"    ENGLISH  GRAMMAR 


BY  THE   LATB 


Professor  W.  D.  WHITNEY, 

Of  Yale  University,  author  of  "  Essentials  of  English  Grammar!^ 

"  Sanskrit  Grammar^''  etc. ;  Editor-in-chief  of 

"  The  Century  Dictionary  "  ; 


Mrs.  S.  E.  H.  LOCKWOOD, 

Formerly  Teacher  of  English  in  the  High  School,  New  Haven,  Conn.^ 
and  author  of^^  Lessons  in  English.'''' 


X2mo.  Cloth.  264  pages.  For  introduction,  70  cents. 

This  sterling  book  was  prepared  under  the  personal  supervision  of 
the  late  Professor  Whitney.  The  formal  statement  of  principles  and 
rules  has  been  modified  only  so  far  as  was  necessary  to  secure  greater 
simplicity.  The  inductive  method  of  the  original  has  been  in  general 
retained.  The  book  may  therefore  be  said  to  bear  the  stamp  of  eminent 
scholarship,  and  to  be  so  far  authoritative. 

Some  special  features  of  the  revision  are  as  follows  : 

Topical  Arrangement.  The  work  has  been  entirely  rewritten  in 
topical  form.  The  improvement  in  typography  will  commend  itself. 
Topical  headings  are  printed  in  boldface  type ;  examples  are  in  smaller 
type,  and  so  arranged  as  to  stand  out  clearly  from  the  text. 

Simplicity  and  Conciseness.  The  aim  has  been  to  make  a  practical, 
rather  than  a  pretentious,  text-book.  To  this  end  the  editor  has  sought 
to  embody  the  most  important  facts  of  the  language,  and  to  set  them 
forth  simply,  distinctly,  and  concisely. 

Abundant  Illustration.  A  conspicuous  feature  of  the  book  is  the 
large  number  of  practical  exercises  for  oral  and  written  work.  More 
than  one  hundred  of  these  are  scattered  through  the  book,  and  there 
are  miscellaneous  exercises  at  the  end  of  each  chapter.  In  selecting 
these  extracts,  the  aim  has  been  to  secure  a  pleasing  and  interesting 
variety  of  such  as  best  illustrate  the  constructions  described  in  the  text. 
As  the  work  of  a  practical  teacher,  it  is  believed  that  these  exercises 
will  greatly  add  to  the  interest  and  profit  of  classes  who  may  use  the 
book. 


GINN   &  COMPANY,  Publishers, 

Boston.  New  York.  Chicago.  Atlanta.  Dallas. 


YB    17250 


5  7 


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